9 billion names of God the integer
This task is a variation of the short story by Arthur C. Clarke.
You are encouraged to solve this task according to the task description, using any language you may know.
(Solvers should be aware of the consequences of completing this task.)
In detail, to specify what is meant by a “name”:
- The integer 1 has 1 name “1”.
- The integer 2 has 2 names “1+1”, and “2”.
- The integer 3 has 3 names “1+1+1”, “2+1”, and “3”.
- The integer 4 has 5 names “1+1+1+1”, “2+1+1”, “2+2”, “3+1”, “4”.
- The integer 5 has 7 names “1+1+1+1+1”, “2+1+1+1”, “2+2+1”, “3+1+1”, “3+2”, “4+1”, “5”.
- Task
Display the first 25 rows of a number triangle which begins:
1
1 1
1 1 1
1 2 1 1
1 2 2 1 1
1 3 3 2 1 1
Where row corresponds to integer , and each column in row from left to right corresponds to the number of names beginning with .
A function should return the sum of the -th row.
Demonstrate this function by displaying: , , , and .
Optionally note that the sum of the -th row is the integer partition function.
Demonstrate this is equivalent to by displaying: , , , and .
- Extra credit
If your environment is able, plot against for .
- Related tasks
11l
<lang 11l>V cache = BigInt(1) F cumu(n)
L(l) :cache.len .. n
V r = [BigInt(0)]
L(x) 1 .. l
r.append(r.last + :cache[l - x][min(x, l - x)])
:cache.append(r)
R :cache[n]
F row(n)
V r = cumu(n) R (0 .< n).map(i -> @r[i + 1] - @r[i])
print(‘rows:’) L(x) 1..10
print(‘#2:’.format(x)‘ ’row(x))
print("\nsums:")
V pp = [BigInt(1)]
F partitions(n)
:pp.append(BigInt(0))
L(k) 1 .. n
V d = n - k * (3 * k - 1) I/ 2
I d < 0
L.break
I k [&] 1 != 0
:pp[n] += :pp[d]
E
:pp[n] -= :pp[d]
d -= k
I d < 0
L.break
I k [&] 1 != 0
:pp[n] += :pp[d]
E
:pp[n] -= :pp[d]
R :pp.last
V ns = Set([23, 123, 1234, 12345]) V max_ns = max(ns)
L(i) 1 .. max_ns
I i > max_ns
L.break
V p = partitions(i)
I i C ns
print(‘#6: #.’.format(i, p))</lang>
- Output:
rows:
1: [1]
2: [1, 1]
3: [1, 1, 1]
4: [1, 2, 1, 1]
5: [1, 2, 2, 1, 1]
6: [1, 3, 3, 2, 1, 1]
7: [1, 3, 4, 3, 2, 1, 1]
8: [1, 4, 5, 5, 3, 2, 1, 1]
9: [1, 4, 7, 6, 5, 3, 2, 1, 1]
10: [1, 5, 8, 9, 7, 5, 3, 2, 1, 1]
sums:
23: 1255
123: 2552338241
1234: 156978797223733228787865722354959930
12345: 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
AArch64 Assembly
<lang AArch64 Assembly> /* ARM assembly AARCH64 Raspberry PI 3B */ /* program integerName64.s */
/*******************************************/ /* Constantes file */ /*******************************************/ /* for this file see task include a file in language AArch64 assembly*/ .include "../includeConstantesARM64.inc"
.equ MAXI, 524
/*********************************/ /* Initialized data */ /*********************************/ .data sMessResult: .asciz "Total : @ pour @ \n" szMessError: .asciz "Number too large !!.\n" szCarriageReturn: .asciz "\n" /*********************************/ /* UnInitialized data */ /*********************************/ .bss sZoneConv: .skip 24 tbNames: .skip 8 * MAXI /*********************************/ /* code section */ /*********************************/ .text .global main main: // entry of program
mov x0,#5 bl functionG mov x0,#23 bl functionG
mov x0,#123 bl functionG mov x0,#524 bl functionG mov x0,#1234 bl functionG
100: // standard end of the program
mov x0, #0 // return code mov x8, #EXIT // request to exit program svc #0 // perform the system call
qAdrszCarriageReturn: .quad szCarriageReturn qAdrsMessResult: .quad sMessResult qAdrtbNames: .quad tbNames qAdrsZoneConv: .quad sZoneConv /******************************************************************/ /* compute function G */ /******************************************************************/ /* x0 contains N */ functionG:
stp x1,lr,[sp,-16]! // save registres stp x2,x3,[sp,-16]! // save registres stp x4,x5,[sp,-16]! // save registres cmp x0,#MAXI + 1 bge 2f mov x3,x0 mov x2,#1
1: // loop compute every item
mov x0,x2 bl computeNumber add x2,x2,#1 cmp x2,x3 ble 1b ldr x1,qAdrsZoneConv // result display bl conversion10 // call decimal conversion ldr x0,qAdrsMessResult ldr x1,qAdrsZoneConv // insert conversion in message bl strInsertAtCharInc mov x4,x0 mov x0,x3 ldr x1,qAdrsZoneConv // result display bl conversion10 // call decimal conversion mov x0,x4 ldr x1,qAdrsZoneConv // insert conversion in message bl strInsertAtCharInc bl affichageMess mov x0,#0 b 100f
2:
ldr x0,qAdrszMessError bl affichageMess mov x0,#-1
100:
ldp x4,x5,[sp],16 // restaur des 2 registres ldp x2,x3,[sp],16 // restaur des 2 registres ldp x1,lr,[sp],16 // restaur des 2 registres ret
qAdrszMessError: .quad szMessError /******************************************************************/ /* random door test strategy */ /******************************************************************/ /* x0 contains N */ computeNumber:
stp x1,lr,[sp,-16]! // save registres stp x2,x3,[sp,-16]! // save registres stp x4,x5,[sp,-16]! // save registres stp x6,x7,[sp,-16]! // save registres ldr x6,qAdrtbNames // table address mov x1,#1 str x1,[x6] // init item 0 mov x1,#0 str x1,[x6,x0,lsl #3] // init item N mov x2,#1 // indice
1:
add x3,x2,x2, lsl #1 sub x4,x3,#1 mul x4,x2,x4 lsr x4,x4,#1 subs x3,x0,x4 // compute new indice blt 90f tst x2,#1 // indice owen ? beq 2f ldr x4,[x6,x3,lsl #3] ldr x5,[x6,x0,lsl #3] add x5,x5,x4 // addition str x5,[x6,x0,lsl #3] b 3f
2: // else substrac
ldr x4,[x6,x3,lsl #3] ldr x5,[x6,x0,lsl #3] sub x5,x5,x4 str x5,[x6,x0,lsl #3]
3:
subs x3,x3,x2 // compute new indice blt 90f tst x2,#1 // owen ? beq 4f ldr x4,[x6,x3,lsl #3] ldr x5,[x6,x0,lsl #3] add x5,x5,x4 str x5,[x6,x0,lsl #3] b 5f
4:
ldr x4,[x6,x3,lsl #3] ldr x5,[x6,x0,lsl #3] sub x5,x5,x4 str x5,[x6,x0,lsl #3]
5:
add x2,x2,#1 cmp x2,x0 ble 1b
90:
ldr x0,[x6,x0,lsl #3] // return last item of table
100:
ldp x6,x7,[sp],16 // restaur des 2 registres ldp x4,x5,[sp],16 // restaur des 2 registres ldp x2,x3,[sp],16 // restaur des 2 registres ldp x1,lr,[sp],16 // restaur des 2 registres ret
/********************************************************/ /* File Include fonctions */ /********************************************************/ /* for this file see task include a file in language AArch64 assembly */ .include "../includeARM64.inc" </lang>
- Output:
Total : 7 pour 5 Total : 1255 pour 23 Total : 2552338241 pour 123 Total : 18324532310194337942 pour 524 Number too large !!.
Ada
<lang Ada>with Ada.Text_IO; with Ada.Numerics.Big_Numbers.Big_Integers;
procedure Names_Of_God is
NN : constant := 100_000; Row_Count : constant := 25; Max_Column : constant := 79;
package Triangle is
procedure Print;
end Triangle;
package Row_Summer is
procedure Calc (N : Integer);
procedure Put_Sums;
end Row_Summer;
package body Row_Summer is
use Ada.Text_IO;
use Ada.Numerics.Big_Numbers.Big_Integers;
P : array (0 .. NN + 1) of Big_Integer := (1, others => 0);
procedure Calc (N : Integer) is
begin
P (N) := 0;
for K in 1 .. N + 1 loop
declare
Add : constant Boolean := K mod 2 /= 0;
D_1 : constant Integer := N - K * (3 * K - 1) / 2;
D_2 : constant Integer := D_1 - K;
begin
exit when D_1 < 0;
if Add
then P (N) := P (N) + P (D_1);
else P (N) := P (N) - P (D_1);
end if;
exit when D_2 < 0;
if Add
then P (N) := P (N) + P (D_2);
else P (N) := P (N) - P (D_2);
end if;
end;
end loop;
end Calc;
procedure Put_Wrapped (Item : Big_Integer) is
Image : constant String := To_String (Item);
begin
Set_Col (11);
for I in Image'Range loop
if Ada.Text_IO.Col >= Max_Column then
Set_Col (12);
end if;
Put (Image (I));
end loop;
end Put_Wrapped;
procedure Put_Sums
is
package Integer_IO is new Ada.Text_IO.Integer_IO (Integer);
Printout : constant array (Natural range <>) of Integer :=
(23, 123, 1234, 12_345, 20_000, 30_000, 40_000, 50_000, NN);
Next : Natural := Printout'First;
begin
for A in 1 .. Printout (Printout'Last) loop
Calc (A);
if A = Printout (Next) then
Put ("G (");
Integer_IO.Put (A, Width => 0);
Put (")");
Put_Wrapped (P (A));
New_Line;
Next := Next + 1;
end if;
end loop;
end Put_Sums;
end Row_Summer;
package body Triangle is
Triangle : array (0 .. Row_Count,
0 .. Row_Count) of Integer := (others => (others => 0));
procedure Calculate is
begin
Triangle (1,1) := 1;
Triangle (2,1) := 1;
Triangle (2,2) := 1;
Triangle (3,1) := 1;
Triangle (3,2) := 1;
Triangle (3,3) := 1;
for Row in 4 .. Row_Count loop
for Col in 1 .. Row loop
if Col * 2 > Row then
Triangle (Row, Col) := Triangle (Row - 1, Col - 1);
else
Triangle (Row, Col) :=
Triangle (Row - 1, Col - 1) +
Triangle (Row - Col, Col);
end if;
end loop;
end loop;
end Calculate;
procedure Print
is
use Ada.Text_IO;
Width : array (1 .. Row_Count) of Natural := (others => 0);
begin
for Row in 1 .. Row_count loop
for Col in 1 .. Row loop
Width (Row) := Width (Row) + Triangle (Row, Col)'Image'Length;
end loop;
end loop;
for Row in 1 .. Row_Count loop
Set_Col (1 + Positive_Count (1 + Width (Width'Last)
- Width (Row)) / 2);
for Col in 1 .. Row loop
Put (Triangle (Row, Col)'Image);
end loop;
New_Line;
end loop;
end Print;
begin
Calculate;
end Triangle;
begin
Triangle.Print; Row_Summer.Put_Sums;
end Names_Of_God;</lang>
- Output:
The program run takes 23 seconds on 2016 MacBook Air.
1
1 1
1 1 1
1 2 1 1
1 2 2 1 1
1 3 3 2 1 1
1 3 4 3 2 1 1
1 4 5 5 3 2 1 1
1 4 7 6 5 3 2 1 1
1 5 8 9 7 5 3 2 1 1
1 5 10 11 10 7 5 3 2 1 1
1 6 12 15 13 11 7 5 3 2 1 1
1 6 14 18 18 14 11 7 5 3 2 1 1
1 7 16 23 23 20 15 11 7 5 3 2 1 1
1 7 19 27 30 26 21 15 11 7 5 3 2 1 1
1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1
1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1
1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1
1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1
1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1
1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1
1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1
1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1
1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1
1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
G (23) 1255
G (123) 2552338241
G (1234) 156978797223733228787865722354959930
G (12345) 6942035795392611681956297720520938446066767309467146362027032170080
6074195845953959951425306140971942519870679768681736
G (20000) 2521148138125296979166195332304704522813289496018115934368503141080
3428442380156495662397073168982436919232478935199490301641182623057
8166735959242113097
G (30000) 4296358424632538517488315748300592091269024864540113906601448061276
4163986215458185192990173314832179564211367228855321718015074490598
095469727784182254987592569621576375743614022636192786
G (40000) 2280772827447072828934057124081695970464622037835161185943949940867
2657828590548093703330014605000554127042566412316061732771683740688
0512642374788938691635864264873546003424774916205066033895952328900
82673857997469797
G (50000) 3626186097141667844592140891595633728165383082527785049015872755414
1099042567120827181227473166105658246308817729102175442616592394326
7067153241385837825618898733387712189158660795738975053844747471259
2979263719012461858719791627302489739548263
G (100000) 2749351056977569651267751632098635268817342931598005475820312598430
2147328114964173055050741660736621590157844774296248940493063070200
4617927644930335101160793424571901557189435097253124661084520063695
5893446424871682878983218234500926285383140459702130713067451062441
9227311238999702284408609370935531629697851569569892196108480158600
569421098519
ARM Assembly
<lang ARM Assembly> /* ARM assembly Raspberry PI */ /* program integerName.s */
/* REMARK 1 : this program use routines in a include file
see task Include a file language arm assembly for the routine affichageMess conversion10 see at end of this program the instruction include */
/* for constantes see task include a file in arm assembly */ /************************************/ /* Constantes */ /************************************/ .include "../constantes.inc"
.equ MAXI, 127
/*********************************/ /* Initialized data */ /*********************************/ .data sMessResult: .asciz "Total : @ \n" szMessError: .asciz "Number too large !!.\n" szCarriageReturn: .asciz "\n" /*********************************/ /* UnInitialized data */ /*********************************/ .bss sZoneConv: .skip 24 tbNames: .skip 4 * MAXI /*********************************/ /* code section */ /*********************************/ .text .global main main: @ entry of program
mov r0,#5 bl functionG mov r0,#23 bl functionG
mov r0,#123 bl functionG mov r0,#1234 bl functionG
100: @ standard end of the program
mov r0, #0 @ return code mov r7, #EXIT @ request to exit program svc #0 @ perform the system call
iAdrszCarriageReturn: .int szCarriageReturn iAdrsMessResult: .int sMessResult iAdrtbNames: .int tbNames iAdrsZoneConv: .int sZoneConv /******************************************************************/ /* compute function G */ /******************************************************************/ /* r0 contains N */ functionG:
push {r1-r3,lr} @ save registers
cmp r0,#MAXI + 1
bge 2f
mov r3,r0
mov r2,#1
1: @ loop compute every item
mov r0,r2 bl computeNumber add r2,r2,#1 cmp r2,r3 ble 1b ldr r1,iAdrsZoneConv @ result display bl conversion10 @ call decimal conversion ldr r0,iAdrsMessResult ldr r1,iAdrsZoneConv @ insert conversion in message bl strInsertAtCharInc bl affichageMess mov r0,#0 b 100f
2:
ldr r0,iAdrszMessError bl affichageMess mov r0,#-1
100:
pop {r1-r3,lr}
bx lr @ return
iAdrszMessError: .int szMessError /******************************************************************/ /* random door test strategy */ /******************************************************************/ /* r0 contains N */ computeNumber:
push {r1-r7,lr} @ save registers
ldr r6,iAdrtbNames @ table address
mov r1,#1
str r1,[r6] @ init item 0
mov r1,#0
str r1,[r6,r0,lsl #2] @ init item N
mov r2,#1 @ indice
1:
add r3,r2,r2, lsl #1 sub r4,r3,#1 mul r4,r2,r4 lsr r4,r4,#1 subs r3,r0,r4 @ compute new indice blt 90f tst r2,#1 @ indice owen ? beq 2f ldr r4,[r6,r3,lsl #2] ldr r5,[r6,r0,lsl #2] add r5,r5,r4 @ addition str r5,[r6,r0,lsl #2] b 3f
2: @ else substrac
ldr r4,[r6,r3,lsl #2] ldr r5,[r6,r0,lsl #2] sub r5,r5,r4 str r5,[r6,r0,lsl #2]
3:
subs r3,r3,r2 @ compute new indice blt 90f tst r2,#1 @ owen ? beq 4f ldr r4,[r6,r3,lsl #2] ldr r5,[r6,r0,lsl #2] add r5,r5,r4 str r5,[r6,r0,lsl #2] b 5f
4:
ldr r4,[r6,r3,lsl #2] ldr r5,[r6,r0,lsl #2] sub r5,r5,r4 str r5,[r6,r0,lsl #2]
5:
add r2,r2,#1 cmp r2,r0 ble 1b
90:
ldr r0,[r6,r0,lsl #2] @ return last item of table
100:
pop {r1-r7,lr}
bx lr @ return
/***************************************************/ /* ROUTINES INCLUDE */ /***************************************************/ .include "../affichage.inc" </lang>
- Output:
Total : 7 Total : 1255 Total : 2552338241 Number too large !!.
AutoHotkey
<lang AutoHotkey>SetBatchLines -1
InputBox, Enter_value, Enter the no. of lines sought array := [] Loop, % 2*Enter_value - 1 Loop, % x := A_Index y := A_Index, Array[x, y] := 1
x := 3
Loop { base_r := x - 1 , x++ , y := 2 , index := x , new := 1
Loop, % base_r - 1 { array[x, new+1] := array[x-1, new] + array[base_r, y] , x++ , new ++ , y++ } x := index If ( mod(x,2) = 0 ) { to_run := floor(x - x/2) , y2 := to_run + 1 } Else { to_run := x - floor(x/2) , y2 := to_run } Loop, % to_run { array[x, y2] := array[x-1, y2-1] , y2++ If ( y2 = Enter_value + 1 ) && ( x = Enter_value ) { Loop, % Enter_value { Loop, % x11 := A_Index { y11 := A_Index , string2 .= " " array[x11, y11] } string2 .= "`n" } MsgBox % string2 ExitApp } } }
~Esc::ExitApp</lang>
- Output:
If user inputs 25, the result shall be:
1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1 1 5 10 11 10 7 5 3 2 1 1 1 6 12 15 13 11 7 5 3 2 1 1 1 6 14 18 18 14 11 7 5 3 2 1 1 1 7 16 23 23 20 15 11 7 5 3 2 1 1 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
C
If we forgo the rows and only want to calculate , using the recurrence relation is a better way. This requires storage for caching instead the -ish for storing all the rows. <lang c>#include <stdio.h>
- include <gmp.h>
- define N 100000
mpz_t p[N + 1];
void calc(int n) { mpz_init_set_ui(p[n], 0);
for (int k = 1; k <= n; k++) { int d = n - k * (3 * k - 1) / 2; if (d < 0) break;
if (k&1)mpz_add(p[n], p[n], p[d]); else mpz_sub(p[n], p[n], p[d]);
d -= k; if (d < 0) break;
if (k&1)mpz_add(p[n], p[n], p[d]); else mpz_sub(p[n], p[n], p[d]); } }
int main(void) { int idx[] = { 23, 123, 1234, 12345, 20000, 30000, 40000, 50000, N, 0 }; int at = 0;
mpz_init_set_ui(p[0], 1);
for (int i = 1; idx[at]; i++) { calc(i); if (i != idx[at]) continue;
gmp_printf("%2d:\t%Zd\n", i, p[i]); at++; } }</lang>
- Output:
23: 1255 123: 2552338241 1234: 156978797223733228787865722354959930 12345: 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736 ...
C#
(this requires a System.Numerics registry reference)
<lang csharp>using System; using System.Collections.Generic; using System.Linq; using System.Numerics;
namespace NamesOfGod {
public class RowSummer
{
const int N = 100000;
public BigInteger[] p;
private void calc(int n)
/* Translated from C */
{
p[n] = 0;
for (int k = 1; k <= n; k++)
{
int d = n - k * (3 * k - 1) / 2;
if (d < 0) break;
if ((k & 1) != 0) p[n] += p[d];
else p[n] -= p[d];
d -= k;
if (d < 0) break;
if ((k & 1) != 0) p[n] += p[d];
else p[n] -= p[d];
}
}
public void PrintSums()
/* translated from C */
{
p = new BigInteger[N + 1];
var idx = new int[] { 23, 123, 1234, 12345, 20000, 30000, 40000, 50000, N, 0 };
int at = 0;
p[0] = 1;
for (int i = 1; idx[at] > 0; i++)
{
calc(i);
if (i != idx[at]) continue;
Console.WriteLine(i + ":\t" + p[i]);
at++;
}
}
}
public class RowPrinter
/* translated from Python */
{
List<List<int>> cache;
public RowPrinter()
{
cache = new List<List<int>> { new List<int> { 1 } };
}
public List<int> cumu(int n)
{
for (int l = cache.Count; l < n + 1; l++)
{
var r = new List<int> { 0 };
for (int x = 1; x < l + 1; x++)
r.Add(r.Last() + cache[l - x][Math.Min(x, l - x)]);
cache.Add(r);
}
return cache[n];
}
public List<int> row(int n)
{
var r = cumu(n);
return (from i in Enumerable.Range(0, n) select r[i + 1] - r[i]).ToList();
}
public void PrintRows()
{
var rows = Enumerable.Range(1, 25).Select(x => string.Join(" ", row(x))).ToList();
var widest = rows.Last().Length;
foreach (var r in rows)
Console.WriteLine(new String(' ', (widest - r.Length) / 2) + r);
}
}
class Program
{
static void Main(string[] args)
{
var rpr = new RowPrinter();
rpr.PrintRows();
var ros = new RowSummer();
ros.PrintSums();
Console.ReadLine();
}
}
} </lang>
- Output:
1
1 1
1 1 1
1 2 1 1
1 2 2 1 1
1 3 3 2 1 1
1 3 4 3 2 1 1
1 4 5 5 3 2 1 1
1 4 7 6 5 3 2 1 1
1 5 8 9 7 5 3 2 1 1
1 5 10 11 10 7 5 3 2 1 1
1 6 12 15 13 11 7 5 3 2 1 1
1 6 14 18 18 14 11 7 5 3 2 1 1
1 7 16 23 23 20 15 11 7 5 3 2 1 1
1 7 19 27 30 26 21 15 11 7 5 3 2 1 1
1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1
1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1
1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1
1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1
1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1
1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1
1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1
1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1
1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1
1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
23: 1255
123: 2552338241
1234: 156978797223733228787865722354959930
12345: 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
20000: 252114813812529697916619533230470452281328949601811593436850314108034284423801564956623970731689824369192324789351994903016411826230578166735959242113097
30000: 42963584246325385174883157483005920912690248645401139066014480612764163986215458185192990173314832179564211367228855321718015074490598095469727784182254987592569621576375743614022636192786
40000: 22807728274470728289340571240816959704646220378351611859439499408672657828590548093703330014605000554127042566412316061732771683740688051264237478893869163586426487354600342477491620506603389595232890082673857997469797
50000: 3626186097141667844592140891595633728165383082527785049015872755414109904256712082718122747316610565824630881772910217544261659239432670671532413858378256188987333877121891586607957389750538447474712592979263719012461858719791627302489739548263
100000: 27493510569775696512677516320986352688173429315980054758203125984302147328114964173055050741660736621590157844774296248940493063070200461792764493033510116079342457190155718943509725312466108452006369558934464248716828789832182345009262853831404597021307130674510624419227311238999702284408609370935531629697851569569892196108480158600569421098519
C++
The Code
see [The Green Triangle]. <lang cpp> // Calculate hypotenuse n of OTT assuming only nothingness, unity, and hyp[n-1] if n>1 // Nigel Galloway, May 6th., 2013
- include <gmpxx.h>
int N{123456}; mpz_class hyp[N-3]; const mpz_class G(const int n,const int g){return g>n?0:(g==1 or n-g<2)?1:hyp[n-g-2];}; void G_hyp(const int n){for(int i=0;i<N-2*n-1;i++) n==1?hyp[n-1+i]=1+G(i+n+1,n+1):hyp[n-1+i]+=G(i+n+1,n+1);} } </lang>
The Alpha and Omega, Beauty
Before displaying the triangle the following code displays hyp as it is transformed by consequtive calls of G_hyp. <lang cpp>
- include <iostream>
- include <iomanip>
int main(){
N=25;
for (int n=1; n<N/2; n++){
G_hyp(n);
for (int g=0; g<N-3; g++) std::cout << std::setw(4) << hyp[g];
std::cout << std::endl;
}
} </lang>
- Output:
2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 2 3 4 5 7 8 10 12 14 16 19 21 24 27 30 33 37 40 44 48 52 12 2 3 5 6 9 11 15 18 23 27 34 39 47 54 64 72 84 94 108 120 52 12 2 3 5 7 10 13 18 23 30 37 47 57 70 84 101 119 141 164 192 120 52 12 2 3 5 7 11 14 20 26 35 44 58 71 90 110 136 163 199 235 192 120 52 12 2 3 5 7 11 15 21 28 38 49 65 82 105 131 164 201 248 235 192 120 52 12 2 3 5 7 11 15 22 29 40 52 70 89 116 146 186 230 248 235 192 120 52 12 2 3 5 7 11 15 22 30 41 54 73 94 123 157 201 230 248 235 192 120 52 12 2 3 5 7 11 15 22 30 42 55 75 97 128 164 201 230 248 235 192 120 52 12 2 3 5 7 11 15 22 30 42 56 76 99 131 164 201 230 248 235 192 120 52 12 2 3 5 7 11 15 22 30 42 56 77 100 131 164 201 230 248 235 192 120 52 12
- The first row is the hypotenuse of the green triangle.
- The second row cols 2 to 21 is the hypotenuse 1 in. Col 1 is the last entry in the horizontal edge of the grey triangle. Col 22 is the first entry of the horizontal edge of the green triangle.
- With subsequent calls the horizontal edges expand until, on the final row, the sequence of hypotenuses is finished and hyp contains the horizontal edge of the OTT.
This must be the most beautiful thing on rosettacode!!! Note that the algorithm requires only this data, and requires only N/2 iterations with the nth iteration performing N-3-2*n calculations.
The One True Triangle, OTT
The following will display OTT(25). <lang cpp> int main(){
N = 25;
std::cout << std::setw(N+52) << "1" << std::endl;
std::cout << std::setw(N+55) << "1 1" << std::endl;
std::cout << std::setw(N+58) << "1 1 1" << std::endl;
std::string ott[N-3];
for (int n=1; n<N/2; n++) {
G_hyp(n);
for (int g=(n-1)*2; g<N-3; g++) {
std::string t = hyp[g-(n-1)].get_str();
//if (t.size()==1) t.insert(t.begin(),1,' ');
ott[g].append(t);
ott[g].append(6-t.size(),' ');
}
}
for(int n = 0; n<N-3; n++) {
std::cout <<std::setw(N+43-3*n) << 1 << " " << ott[n];
for (int g = (n+1)/2; g>0; g--) {
std::string t{hyp[g-1].get_str()};
t.append(6-t.size(),' ');
std::cout << t;
}
std::cout << "1 1" << std::endl;
}
</lang>
- Output:
1
1 1
1 1 1
1 2 1 1
1 2 2 1 1
1 3 3 2 1 1
1 3 4 3 2 1 1
1 4 5 5 3 2 1 1
1 4 7 6 5 3 2 1 1
1 5 8 9 7 5 3 2 1 1
1 5 10 11 10 7 5 3 2 1 1
1 6 12 15 13 11 7 5 3 2 1 1
1 6 14 18 18 14 11 7 5 3 2 1 1
1 7 16 23 23 20 15 11 7 5 3 2 1 1
1 7 19 27 30 26 21 15 11 7 5 3 2 1 1
1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1
1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1
1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1
1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1
1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1
1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1
1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1
1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1
1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1
1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
Values of Integer Partition Function
Values of the Integer Partition function may be extracted as follows: <lang cpp>
- include <iostream>
int main(){
for (int n=1; n<N/2; n++) G_hyp(n);
std::cout << "G(23) = " << hyp[21] << std::endl;
std::cout << "G(123) = " << hyp[121] << std::endl;
std::cout << "G(1234) = " << hyp[1232] << std::endl;
std::cout << "G(12345) = " << hyp[12343] << std::endl;
mpz_class r{3};
for (int i = 0; i<N-3; i++) r += hyp[i];
std::cout << "G(123456) = " << r << std::endl;
} </lang>
- Output:
G(23) = 1255 G(123) = 2552338241 G(1234) = 156978797223733228787865722354959930 G(12345) = 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736 G(123456) = 30817659578536496678545317146533980855296613274507139217608776782063054452191537379312358383342446230621170608408020911309259407611257151683372221925128388387168451943800027128045369650890220060901494540459081545445020808726917371699102825508039173543836338081612528477859613355349851184591540231790254269948278726548570660145691076819912972162262902150886818986555127204165221706149989
Clojure
<lang clojure>(defn nine-billion-names [row column]
(cond (<= row 0) 0
(<= column 0) 0
(< row column) 0
(= row 1) 1
:else (let [addend (nine-billion-names (dec row) (dec column))
augend (nine-billion-names (- row column) column)]
(+ addend augend))))
(defn print-row [row]
(doseq [x (range 1 (inc row))] (print (nine-billion-names row x) \space)) (println))
(defn print-triangle [rows]
(doseq [x (range 1 (inc rows))] (print-row x)))
(print-triangle 25)</lang>
Common Lisp
<lang lisp>(defun 9-billion-names (row column)
(cond ((<= row 0) 0)
((<= column 0) 0)
((< row column) 0) ((equal row 1) 1) (t (let ((addend (9-billion-names (1- row) (1- column))) (augend (9-billion-names (- row column) column))) (+ addend augend)))))
(defun 9-billion-names-triangle (rows)
(loop for row from 1 to rows
collect (loop for column from 1 to row
collect (9-billion-names row column))))
(9-billion-names-triangle 25) </lang>
Crystal
Naive Solution
<lang ruby>def g(n, g)
return 1 unless 1 < g && g < n-1
(2..g).reduce(1){ |res, q| res + (q > n-g ? 0 : g(n-g, q)) }
end
(1..25).each { |n| puts (1..n).map { |g| "%4s" % g(n, g) }.join }</lang>
- Output:
1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1 1 5 10 11 10 7 5 3 2 1 1 1 6 12 15 13 11 7 5 3 2 1 1 1 6 14 18 18 14 11 7 5 3 2 1 1 1 7 16 23 23 20 15 11 7 5 3 2 1 1 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
D
Producing rows
<lang d>import std.stdio, std.bigint, std.algorithm, std.range;
auto cumu(in uint n) {
__gshared cache = 1.BigInt; foreach (l; cache.length .. n + 1) { auto r = [0.BigInt]; foreach (x; 1 .. l + 1) r ~= r.back + cache[l - x][min(x, l - x)]; cache ~= r; } return cache[n];
}
auto row(in uint n) {
auto r = n.cumu; return n.iota.map!(i => r[i + 1] - r[i]);
}
void main() {
writeln("Rows:");
foreach (x; 1 .. 11)
writefln("%2d: %s", x, x.row);
writeln("\nSums:");
foreach (x; [23, 123, 1234])
writeln(x, " ", x.cumu.back);
}</lang>
- Output:
Rows: 1: [1] 2: [1, 1] 3: [1, 1, 1] 4: [1, 2, 1, 1] 5: [1, 2, 2, 1, 1] 6: [1, 3, 3, 2, 1, 1] 7: [1, 3, 4, 3, 2, 1, 1] 8: [1, 4, 5, 5, 3, 2, 1, 1] 9: [1, 4, 7, 6, 5, 3, 2, 1, 1] 10: [1, 5, 8, 9, 7, 5, 3, 2, 1, 1] Sums: 23 1255 123 2552338241 1234 156978797223733228787865722354959930
Only partition functions
<lang d>import std.stdio, std.bigint, std.algorithm;
struct Names {
BigInt[] p = [1.BigInt];
int opApply(int delegate(ref immutable int, ref BigInt) dg) {
int result;
foreach (immutable n; 1 .. int.max) {
p.assumeSafeAppend;
p ~= 0.BigInt;
foreach (immutable k; 1 .. n + 1) {
auto d = n - k * (3 * k - 1) / 2;
if (d < 0)
break;
if (k & 1)
p[n] += p[d];
else
p[n] -= p[d];
d -= k;
if (d < 0)
break;
if (k & 1)
p[n] += p[d];
else
p[n] -= p[d];
}
result = dg(n, p[n]);
if (result) break;
}
return result; }
}
void main() {
immutable ns = [23:0, 123:0, 1234:0, 12345:0]; immutable maxNs = ns.byKey.reduce!max;
foreach (immutable i, p; Names()) {
if (i > maxNs)
break;
if (i in ns)
writefln("%6d: %s", i, p);
}
}</lang>
- Output:
23: 1255 123: 2552338241 1234: 156978797223733228787865722354959930 12345: 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
- Output:
for a larger input, with newlines
123456: 3081765957853649667854531714653398085529661327450713921760877 6782063054452191537379312358383342446230621170608408020911309 2594076112571516833722219251283883871684519438000271280453696 5089022006090149454045908154544502080872691737169910282550803 9173543836338081612528477859613355349851184591540231790254269 9482787265485706601456910768199129721622629021508868189865551 27204165221706149989
Runtime up to 123456: about 56 seconds (about 50 with ldc2) because currently std.bigint is not fast.
Dart
<lang Dart>import 'dart:math';
List<BigInt> partitions(int n) {
var cache = List<List<BigInt>>.filled(1, List<BigInt>.filled(1, BigInt.from(1)), growable: true);
for(int length = cache.length; length < n + 1; length++) {
var row = List<BigInt>.filled(1, BigInt.from(0), growable: true);
for(int index = 1; index < length + 1; index++) {
var partAtIndex = row[row.length - 1] + cache[length - index][min(index, length - index)];
row.add(partAtIndex);
}
cache.add(row);
}
return cache[n];
}
List<BigInt> row(int n) {
var parts = partitions(n); return List<BigInt>.generate(n, (int index) => parts[index + 1] - parts[index]);
}
void printRows({int min = 1, int max = 11}) {
int maxDigits = max.toString().length;
print('Rows:');
for(int i in List.generate(max - min, (int index) => index + min)) {
print((' ' * (maxDigits - i.toString().length)) + '$i: ${row(i)}');
}
}
void printSums(List<int> args) {
print('Sums:');
for(int i in args) {
print('$i: ${partitions(i)[i]}');
}
}</lang>
In main: <lang Dart> import 'package:DD1_NamesOfGod/DD1_NamesOfGod.dart' as names_of_god;
main(List<String> arguments) {
names_of_god.printRows(min: 1, max: 11); names_of_god.printSums([23, 123, 1234, 12345]);
}</lang>
- Output:
Rows: 1: [1] 2: [1, 1] 3: [1, 1, 1] 4: [1, 2, 1, 1] 5: [1, 2, 2, 1, 1] 6: [1, 3, 3, 2, 1, 1] 7: [1, 3, 4, 3, 2, 1, 1] 8: [1, 4, 5, 5, 3, 2, 1, 1] 9: [1, 4, 7, 6, 5, 3, 2, 1, 1] 10: [1, 5, 8, 9, 7, 5, 3, 2, 1, 1] Sums: 23: 1255 123: 2552338241 1234: 156978797223733228787865722354959930 12345: 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
Dyalect
<lang Dyalect>var cache = 1
func namesOfGod(n) {
for l in cache.len()..n {
var r = [0]
if l == 1 {
r.add(r[r.len() - 1] + cache[0][0])
} else {
for x in 1..l {
r.add(r[r.len() - 1] + cache[l - x][min(x, l-x)])
}
}
cache.add(r)
}
return cache[n]
}
func row(n) {
let r = namesOfGod(n)
var returnArray = []
for i in 0..<n {
returnArray.add(r[i + 1] - r[i])
}
return returnArray
}
for x in 1..25 {
print("\(x): \(row(x))")
}</lang>
Output:
1: [1] 2: [1, 1] 3: [1, 1, 1] 4: [1, 2, 1, 1] 5: [1, 2, 2, 1, 1] 6: [1, 3, 3, 2, 1, 1] 7: [1, 3, 4, 3, 2, 1, 1] 8: [1, 4, 5, 5, 3, 2, 1, 1] 9: [1, 4, 7, 6, 5, 3, 2, 1, 1] 10: [1, 5, 8, 9, 7, 5, 3, 2, 1, 1] 11: [1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1] 12: [1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1] 13: [1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1] 14: [1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1, 1] 15: [1, 7, 19, 27, 30, 26, 21, 15, 11, 7, 5, 3, 2, 1, 1] 16: [1, 8, 21, 34, 37, 35, 28, 22, 15, 11, 7, 5, 3, 2, 1, 1] 17: [1, 8, 24, 39, 47, 44, 38, 29, 22, 15, 11, 7, 5, 3, 2, 1, 1] 18: [1, 9, 27, 47, 57, 58, 49, 40, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 19: [1, 9, 30, 54, 70, 71, 65, 52, 41, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 20: [1, 10, 33, 64, 84, 90, 82, 70, 54, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 21: [1, 10, 37, 72, 101, 110, 105, 89, 73, 55, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 22: [1, 11, 40, 84, 119, 136, 131, 116, 94, 75, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 23: [1, 11, 44, 94, 141, 163, 164, 146, 123, 97, 76, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 24: [1, 12, 48, 108, 164, 199, 201, 186, 157, 128, 99, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 25: [1, 12, 52, 120, 192, 235, 248, 230, 201, 164, 131, 100, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
Elixir
Naive Solution <lang elixir>defmodule God do
def g(n,g) when g == 1 or n < g, do: 1
def g(n,g) do
Enum.reduce(2..g, 1, fn q,res ->
res + (if q > n-g, do: 0, else: g(n-g,q))
end)
end
end
Enum.each(1..25, fn n ->
IO.puts Enum.map(1..n, fn g -> "#{God.g(n,g)} " end)
end)</lang>
- Output:
1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1 1 5 10 11 10 7 5 3 2 1 1 1 6 12 15 13 11 7 5 3 2 1 1 1 6 14 18 18 14 11 7 5 3 2 1 1 1 7 16 23 23 20 15 11 7 5 3 2 1 1 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
Erlang
Step 1: Print the pyramid for a smallish number of names. The P function is implement as described on partition function, (see 59 on that page). This is slow for N > 100, but works fine for the example: 10. <lang Erlang> -module(triangle). -export([start/1]). start(N)->
print(1,1,N).
print(N,N,N)-> 1; print(A,B,N) when A>=B->
io:format("~p ",[formula(A,B)]),
print(A,B+1,N);
print(A,B,N) when B>A->
io:format("~n"),
print(A+1,1,N).
formula(_,0)->
0;
formula(B,B)->
1;
formula(A,B) when B>A->
0;
formula(A1,B1)->
formula(A1-1,B1-1)+formula(A1-B1,B1).
</lang>
- Output:
If user inputs 25, the result shall be:
1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1 1 5 10 11 10 7 5 3 2 1 1 1 6 12 15 13 11 7 5 3 2 1 1 1 6 14 18 18 14 11 7 5 3 2 1 1 1 7 16 23 23 20 15 11 7 5 3 2 1 1 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
Factor
<lang factor>USING: combinators io kernel math math.ranges memoize prettyprint sequences ;
MEMO: p ( m n -- o )
{
{ [ dup zero? ] [ nip ] }
{ [ 2dup = ] [ 2drop 1 ] }
{ [ 2dup < ] [ 2drop 0 ] }
[ [ [ 1 - ] bi@ p ] [ [ - ] [ ] bi p + ] 2bi ]
} cond ;
- row ( n -- seq ) dup [1,b] [ p ] with map ;
- .row ( n -- ) row [ pprint bl ] each nl ;
- .triangle ( n -- ) [1,b] [ .row ] each ;
- G ( n -- sum ) row sum ;
25 .triangle nl "Sums:" print { 23 123 1234 12345 } [ dup pprint bl G . ] each</lang>
- Output:
1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1 1 5 10 11 10 7 5 3 2 1 1 1 6 12 15 13 11 7 5 3 2 1 1 1 6 14 18 18 14 11 7 5 3 2 1 1 1 7 16 23 23 20 15 11 7 5 3 2 1 1 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1 Sums: 23 1255 123 2552338241 1234 156978797223733228787865722354959930 ^C
Forth
NEEDS -xopg ANEW -nbnog \ The Nine Billion Names of God .arbitrary.p #100000 =: N CREATE idx[ #23 , #123 , #1234 , #12345 , #20000 , #30000 , #40000 , #50000 , N , 0 , N GARRAY p : CALC ( n -- ) 0 LOCALS| d n | n 1+ 1 ?DO I 3 * 1- I 2 */ n SWAP - TO d d 0< ?LEAVE I 1 AND IF LET p[n]=p[n]+p[d]: ELSE LET p[n]=p[n]-p[d]: ENDIF I -TO d d 0< ?LEAVE I 1 AND IF LET p[n]=p[n]+p[d]: ELSE LET p[n]=p[n]-p[d]: ENDIF LOOP ; : .GOD ( -- ) 0 LOCAL at LET p[0]=1: N 1 DO I CALC idx[ at CELL[] @ I = IF CR I 5 .R ." : " LET. p[I]: 1 +TO at ENDIF LOOP ; : .ABOUT ( -- ) ." Try: .GOD" ;
- Output:
FORTH> .god 23: 1.2550000000000000000000000000000000000000e+0003 123: 2.5523382410000000000000000000000000000000e+0009 1234: 1.5697879722373322878786572235495993000000e+0035 12345: 6.9420357953926116819562977205209384460667e+0118 20000: 2.5211481381252969791661953323047045228132e+0152 30000: 4.2963584246325385174883157483005920912690e+0187 40000: 2.2807728274470728289340571240816959704646e+0217 50000: 3.6261860971416678445921408915956337281653e+0243 ok
FreeBASIC
<lang freebasic>' version 03-11-2016 ' compile with: fbc -s console
- Include Once "gmp.bi"
Sub partitions(max As ULong, p() As MpZ_ptr)
' based on Numericana code example Dim As ULong a, b, i, k Dim As Long j
Dim As Mpz_ptr s = Allocate(Len(__mpz_struct)) : Mpz_init(s)
Mpz_set_ui(p(0), 1)
For i = 1 To max
j = 1 : k = 1 : b = 2 : a = 5
While j > 0
' j = i - (3*k*k+k) \ 2
j = i - b : b = b + a : a = a + 3
If j >= 0 Then
If k And 1 Then Mpz_add(s, s, p(j)) Else Mpz_sub(s, s, p(j))
End If
j = j + k
If j >= 0 Then
If k And 1 Then Mpz_add(s, s, p(j)) Else Mpz_sub(s, s, p(j))
End If
k = k +1
Wend
Mpz_swap(p(i), s)
Next
Mpz_clear(s)
End Sub
' ------=< MAIN >=------
Dim As ULong n, k, max = 25 ' with max > 416 the numbers become Dim As ULongInt p(max, max) ' to big for a 64bit unsigned integer
p(1, 1) = 1 ' fill the first 3 rows p(2, 1) = 1 : p(2, 2) = 1 p(3, 1) = 1 : p(3, 2) = 1 : p(3, 3) = 1
For n = 4 To max ' fill the rest
For k = 1 To n
If k * 2 > n Then
p(n,k)= p(n-1,k-1)
Else
p(n,k) = p(n-1,k-1) + p(n-k, k)
End If
Next
Next
For n = 1 To 25 ' print the triangle
Print Space((max - n) * 2);
For k = 1 To n
Print Using "####"; p(n, k);
Next
Print
Next Print : print
' calculate the integer partition
max = 123456 ' 1234567 takes about ten minutes Dim As ZString Ptr ans
ReDim big_p(max) As Mpz_ptr For n = 0 To max
big_p(n) = Allocate(Len(__mpz_struct)) : Mpz_init(big_p(n))
Next
partitions(max, big_p())
For n = 1 To Len(Str(max))
k = Val(Left(Str(max), n))
ans = Mpz_get_str (0, 10, big_p(k))
Print Space(10 - n); "P("; Str(k); ") = "; *ans
Next
For n = 0 To max
Mpz_clear(big_p(n))
Next
' empty keyboard buffer While InKey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>
- Output:
1
1 1
1 1 1
1 2 1 1
1 2 2 1 1
1 3 3 2 1 1
1 3 4 3 2 1 1
1 4 5 5 3 2 1 1
1 4 7 6 5 3 2 1 1
1 5 8 9 7 5 3 2 1 1
1 5 10 11 10 7 5 3 2 1 1
1 6 12 15 13 11 7 5 3 2 1 1
1 6 14 18 18 14 11 7 5 3 2 1 1
1 7 16 23 23 20 15 11 7 5 3 2 1 1
1 7 19 27 30 26 21 15 11 7 5 3 2 1 1
1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1
1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1
1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1
1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1
1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1
1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1
1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1
1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1
1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1
1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
P(1) = 1
P(12) = 77
P(123) = 2552338241
P(1234) = 156978797223733228787865722354959930
P(12345) = 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
P(123456) = 30817659578536496678545317146533980855296613274507139217608776782063054452191537379312358383342446230621170608408020911309259407611257151683372221925128388387168451943800027128045369650890220060901494540459081545445020808726917371699102825508039173543836338081612528477859613355349851184591540231790254269948278726548570660145691076819912972162262902150886818986555127204165221706149989
Frink
This demonstrates using a class that memoizes results to improve efficiency and reduce later calculation. It verifies its results against Frink's built-in and much more memory-and-space-efficient partitionCount function which uses Euler's pentagonal method for counting partitions.
<lang frink> class PartitionCount {
// Array of elements class var triangle = [[0],[0,1]]
// Array of cumulative sums in each row. class var sumTriangle = [[1],[0,1]]
class calcRowsTo[toRow] :=
{
for row = length[triangle] to toRow
{
triangle@row = workrow = new array[[row+1],0]
sumTriangle@row = sumworkrow = new array[[row+1],0]
oversum = 0
for col = 1 to row
{
otherRow = row-col
sum = sumTriangle@otherRow@min[col,otherRow]
workrow@col = sum
oversum = oversum + sum
sumworkrow@col = oversum
}
}
}
class rowSum[row] :=
{
calcRowsTo[row]
return sumTriangle@row@row
}
}
PartitionCount.calcRowsTo[25] for row=1 to 25 {
for col=1 to row
print[PartitionCount.triangle@row@col + " "]
println[]
}
// Test against Frink's built-in much faster partitionCount function that uses // Euler's pentagonal method for counting partitions. testRow[row] := {
sum = PartitionCount.rowSum[row] println["$row\t$sum\t" + (sum == partitionCount[row] ? "correct" : "incorrect")]
}
println[] testRow[23] testRow[123] testRow[1234] testRow[12345] </lang>
1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1 1 5 10 11 10 7 5 3 2 1 1 1 6 12 15 13 11 7 5 3 2 1 1 1 6 14 18 18 14 11 7 5 3 2 1 1 1 7 16 23 23 20 15 11 7 5 3 2 1 1 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1 23 1255 correct 123 2552338241 correct 1234 156978797223733228787865722354959930 correct 12345 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736 correct
GAP
The partition function is built-in. <lang gap>PrintArray(List([1 .. 25], n -> List([1 .. n], k -> NrPartitions(n, k))));
[ [ 1 ],
[ 1, 1 ], [ 1, 1, 1 ], [ 1, 2, 1, 1 ], [ 1, 2, 2, 1, 1 ], [ 1, 3, 3, 2, 1, 1 ], [ 1, 3, 4, 3, 2, 1, 1 ], [ 1, 4, 5, 5, 3, 2, 1, 1 ], [ 1, 4, 7, 6, 5, 3, 2, 1, 1 ], [ 1, 5, 8, 9, 7, 5, 3, 2, 1, 1 ], [ 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1 ], [ 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 7, 19, 27, 30, 26, 21, 15, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 8, 21, 34, 37, 35, 28, 22, 15, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 8, 24, 39, 47, 44, 38, 29, 22, 15, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 9, 27, 47, 57, 58, 49, 40, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 9, 30, 54, 70, 71, 65, 52, 41, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 10, 33, 64, 84, 90, 82, 70, 54, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 10, 37, 72, 101, 110, 105, 89, 73, 55, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 11, 40, 84, 119, 136, 131, 116, 94, 75, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 11, 44, 94, 141, 163, 164, 146, 123, 97, 76, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 12, 48, 108, 164, 199, 201, 186, 157, 128, 99, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1 ], [ 1, 12, 52, 120, 192, 235, 248, 230, 201, 164, 131, 100, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1 ] ]
List([23, 123, 1234, 12345], NrPartitions);
[ 1255, 2552338241, 156978797223733228787865722354959930,
69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736 ]</lang>
Go
<lang go>package main
import ( "fmt" "math/big" )
func main() {
intMin := func(a, b int) int { if a < b { return a } else { return b } }
var cache = [][]*big.IntTemplate:Big.NewInt(1)
cumu := func(n int) []*big.Int { for y := len(cache); y <= n; y++ { row := []*big.Int{big.NewInt(0)} for x := 1; x <= y; x++ { cacheValue := cache[y-x][intMin(x, y-x)] row = append(row, big.NewInt(0).Add(row[len(row)-1], cacheValue)) } cache = append(cache, row) } return cache[n] }
row := func(n int) { e := cumu(n) for i := 0; i < n; i++ { fmt.Printf(" %v ", (big.NewInt(0).Sub(e[i+1], e[i])).Text(10)) } fmt.Println() }
fmt.Println("rows:") for x := 1; x < 11; x++ { row(x) } fmt.Println()
fmt.Println("sums:") for _, num := range [...]int{23, 123, 1234, 12345} { r := cumu(num) fmt.Printf("%d %v\n", num, r[len(r)-1].Text(10)) } }</lang>
- Output:
rows: 1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1 sums: 23 1255 123 2552338241 1234 156978797223733228787865722354959930 12345 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
Groovy
<lang groovy> def partitions(c) {
def p=[];
int k = 0;
p[k] = c;
int counter=0;
def counts=[];
for (i in 0..c-1) {counts[i]=0;}
while (true)
{
counter++;
counts[p[0]-1]=counts[p[0]-1]+1; int rem_val = 0;
while (k >= 0 && p[k] == 1)
{ rem_val += p[k];
k--;}
if (k < 0) { break;}
p[k]--;
rem_val++;
while (rem_val > p[k])
{
p[k+1] = p[k];
rem_val = rem_val - p[k];
k++;
}
p[k+1] = rem_val;
k++;
}
println counts; return counter; }
static void main(String[] args)
{
for( i in 1..25 )
{partitions(i);}
}
</lang>
- Output:
[1] [1, 1] [1, 1, 1] [1, 2, 1, 1] [1, 2, 2, 1, 1] [1, 3, 3, 2, 1, 1] [1, 3, 4, 3, 2, 1, 1] [1, 4, 5, 5, 3, 2, 1, 1] [1, 4, 7, 6, 5, 3, 2, 1, 1] [1, 5, 8, 9, 7, 5, 3, 2, 1, 1] [1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1] [1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1] [1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1] [1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1, 1] [1, 7, 19, 27, 30, 26, 21, 15, 11, 7, 5, 3, 2, 1, 1] [1, 8, 21, 34, 37, 35, 28, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 8, 24, 39, 47, 44, 38, 29, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 9, 27, 47, 57, 58, 49, 40, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 9, 30, 54, 70, 71, 65, 52, 41, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 10, 33, 64, 84, 90, 82, 70, 54, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 10, 37, 72, 101, 110, 105, 89, 73, 55, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 11, 40, 84, 119, 136, 131, 116, 94, 75, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 11, 44, 94, 141, 163, 164, 146, 123, 97, 76, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 12, 48, 108, 164, 199, 201, 186, 157, 128, 99, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 12, 52, 120, 192, 235, 248, 230, 201, 164, 131, 100, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
Haskell
<lang haskell>import Data.List (mapAccumL)
cumu :: Integer cumu = [1] : map (scanl (+) 0) rows
rows :: Integer rows = snd $ mapAccumL f [] cumu where
f r row = (rr, new_row) where
new_row = map head rr
rr = map tailKeepOne (row:r)
tailKeepOne [x] = [x]
tailKeepOne (_:xs) = xs
sums n = cumu !! n !! n --curiously, the following seems to be faster --sums = sum . (rows!!)
main :: IO () main = do
mapM_ print $ take 10 rows mapM_ (print.sums) [23, 123, 1234, 12345]</lang>
- Output:
[1] [1,1] [1,1,1] [1,2,1,1] [1,2,2,1,1] [1,3,3,2,1,1] [1,3,4,3,2,1,1] [1,4,5,5,3,2,1,1] [1,4,7,6,5,3,2,1,1] [1,5,8,9,7,5,3,2,1,1] 1255 2552338241 156978797223733228787865722354959930 ^C (probably don't have enough memory for 12345 anyway)
Icon and Unicon
This is a Unicon-specific solution.
<lang unicon>procedure main(A)
n := integer(!A) | 10 every r := 2 to (n+1) do write(right(r-1,2),": ",showList(row(r))) write() every r := 23 | 123 | 1234 | 12345 do write(r," ",cumu(r+1)[-1])
end
procedure cumu(n)
static cache initial cache := 1 every l := *cache to n do { every (r := [0], x := !l) do put(r, r[-1]+cache[1+l-x][1+min(x,l-x)]) put(cache, r) } return cache[n]
end
procedure row(n)
return (r := cumu(n), [: (i := !(*r-1), r[i+1]-r[i]) :]) | r
end
procedure showList(A)
every (s := "[") ||:= (!A||", ") return s[1:-2]||"]"
end</lang>
- Output:
(terminated without waiting for output of cumu(12345))
->9bnogti 1: [1] 2: [1, 1] 3: [1, 1, 1] 4: [1, 2, 1, 1] 5: [1, 2, 2, 1, 1] 6: [1, 3, 3, 2, 1, 1] 7: [1, 3, 4, 3, 2, 1, 1] 8: [1, 4, 5, 5, 3, 2, 1, 1] 9: [1, 4, 7, 6, 5, 3, 2, 1, 1] 10: [1, 5, 8, 9, 7, 5, 3, 2, 1, 1] 23 1255 123 2552338241 1234 156978797223733228787865722354959930 ^C ->
J
Recursive calculation of a row element: <lang j>T=: 0:`1:`(($:&<:+ - $: ])`0:@.(0=]))@.(1+*@-) M. "0</lang> Calculation of the triangle: <lang j>rows=: <@(#~0<])@({: T ])\@i.</lang> Show triangle: <lang j> ({.~1+1 i:~ '1'=])"1 ":> }.rows 1+10 1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1</lang>
Note that we've gone to extra work, here, in this show triangle example, to keep columns aligned when we have multi-digit values. But then we limited the result to one digit values because that is prettier.
Calculate row sums: <lang j>rowSums=: 3 :0"0
z=. (y+1){. 1x
for_ks. <\1+i.y do.
n=.{: k=.>ks
r=.#c=. ({.~* i._1:)(n,0.5 _1.5) p. k
s=.#d=.({.~* i._1:)c-r{.k
'v i'=.|: \:~(c,d),. r ,&({.&k) s
a=. +/(n{z),(_1^1x+2|i) * v{z
z=. a n}z
end.
)</lang>
- Output:
({ [: rowSums >./) 3 23 123 1234
3 1255 2552338241 156978797223733228787865722354959930
Java
<lang java>import java.math.BigInteger; import java.util.*; import static java.util.Arrays.asList; import static java.util.stream.Collectors.toList; import static java.util.stream.IntStream.range; import static java.lang.Math.min;
public class Test {
static List<BigInteger> cumu(int n) {
List<List<BigInteger>> cache = new ArrayList<>();
cache.add(asList(BigInteger.ONE));
for (int L = cache.size(); L < n + 1; L++) {
List<BigInteger> r = new ArrayList<>();
r.add(BigInteger.ZERO);
for (int x = 1; x < L + 1; x++)
r.add(r.get(r.size() - 1).add(cache.get(L - x).get(min(x, L - x))));
cache.add(r);
}
return cache.get(n);
}
static List<BigInteger> row(int n) {
List<BigInteger> r = cumu(n);
return range(0, n).mapToObj(i -> r.get(i + 1).subtract(r.get(i)))
.collect(toList());
}
public static void main(String[] args) {
System.out.println("Rows:");
for (int x = 1; x < 11; x++)
System.out.printf("%2d: %s%n", x, row(x));
System.out.println("\nSums:");
for (int x : new int[]{23, 123, 1234}) {
List<BigInteger> c = cumu(x);
System.out.printf("%s %s%n", x, c.get(c.size() - 1));
}
}
}</lang>
Rows: 1: [1] 2: [1, 1] 3: [1, 1, 1] 4: [1, 2, 1, 1] 5: [1, 2, 2, 1, 1] 6: [1, 3, 3, 2, 1, 1] 7: [1, 3, 4, 3, 2, 1, 1] 8: [1, 4, 5, 5, 3, 2, 1, 1] 9: [1, 4, 7, 6, 5, 3, 2, 1, 1] 10: [1, 5, 8, 9, 7, 5, 3, 2, 1, 1] Sums: 23 1255 123 2552338241 1234 156978797223733228787865722354959930
JavaScript
Solution 1
<lang JavaScript> (function () {
var cache = [
[1]
];
//this was never needed.
/* function PyRange(start, end, step) {
step = step || 1;
if (!end) {
end = start;
start = 0;
}
var arr = [];
for (var i = start; i < end; i += step) arr.push(i);
return arr;
}*/
function cumu(n) {
var /*ra = PyRange(cache.length, n + 1),*/ //Seems there is a better version for this
r, l, x, Aa, Mi;
// for (ll in ra) { too pythony
for (l=cache.length;l<n+1;l++) {
r = [0];
// l = ra[ll]; // ran = PyRange(1, l + 1); // for (xx in ran) {
for(x=1;x<l+1;x++){
// x = ran[xx];
r.push(r[r.length - 1] + (Aa = cache[l - x < 0 ? cache.length - (l - x) : l - x])[(Mi = Math.min(x, l - x)) < 0 ? Aa.length - Mi : Mi]);
}
cache.push(r);
}
return cache[n];
}
function row(n) {
var r = cumu(n),
// rra = PyRange(n),
leArray = [],
i;
// for (ii in rra) {
for (i=0;i<n;i++) {
// i = rra[ii];
leArray.push(r[i + 1] - r[i]);
}
return leArray;
}
console.log("Rows:");
for (iterator = 1; iterator < 12; iterator++) {
console.log(row(iterator));
}
// PL clearly this was not tested: // console.log("Sums")[23, 123, 1234, 12345].foreach(function (a) {
console.log("Sums");
[23, 123, 1234, 12345].forEach(function (a) {
var s = cumu(a);
console.log(a, s[s.length - 1]);
});
})() </lang>
Solution 2
Clean and straightforward solution <lang javascript> function genTriangle(n){ // O(n^3) time and O(n^2) space
var a = new Array(n)
for (let i = 0; i < n; i++){
a[i] = new Array(i+1)
for (let j = 0; j < i; j++){
a[i][j] = 0
let s = i-j-1, k = Math.min(s, j)
while (k >= 0) a[i][j] += a[s][k--]
}
a[i][i] = 1
}
return a.map(x => x.join(" ")).join("\n")
}
function G(n){ // At least O(n^2) time and O(n) space
var a = new Array(n+1)
a[0] = 1n
for (let i = 1; i <= n; i++){
a[i] = 0n
for (let k = 1, s = 1; s <= i;){
a[i] += (k & 1 ? a[i-s]:-a[i-s])
k > 0 ? (s += k, k = -k):(k = -k+1, s = k*(3*k-1)/2)
}
}
return a[n]
}
console.log(genTriangle(25)) console.log("")
for (const x of [23, 123, 1234, 12345]){
console.log("G(" + x + ") = " + G(x))
} </lang>
- Output:
1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1 1 5 10 11 10 7 5 3 2 1 1 1 6 12 15 13 11 7 5 3 2 1 1 1 6 14 18 18 14 11 7 5 3 2 1 1 1 7 16 23 23 20 15 11 7 5 3 2 1 1 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1 G(23) = 1255 G(123) = 2552338241 G(1234) = 156978797223733228787865722354959930 G(12345) = 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
Julia
<lang julia> using Combinatorics, StatsBase
namesofline(n) = counts([x[1] for x in integer_partitions(n)])
function centerjustpyramid(n)
maxwidth = length(string(namesofline(n)))
for i in 1:n
s = string(namesofline(i))
println(" " ^ div(maxwidth - length(s), 2), s)
end
end
centerjustpyramid(25)
const cachecountpartitions = Dict{BigInt,BigInt}() function countpartitions(n::BigInt)
if n < 0
0
elseif n < 2
1
elseif (np = get(cachecountpartitions, n, 0)) > 0
np
else
np = 0
sgn = 1
for k = 1:n
np += sgn * (countpartitions(n - (k*(3k-1)) >> 1) + countpartitions(n - (k*(3k+1)) >> 1))
sgn = -sgn
end
cachecountpartitions[n] = np
end
end
G(n) = countpartitions(BigInt(n))
for g in [23, 123, 1234, 12345]
@time println("\nG($g) is $(G(g))")
end
</lang>
- Output:
[1] [1, 1] [1, 1, 1] [1, 2, 1, 1] [1, 2, 2, 1, 1] [1, 3, 3, 2, 1, 1] [1, 3, 4, 3, 2, 1, 1] [1, 4, 5, 5, 3, 2, 1, 1] [1, 4, 7, 6, 5, 3, 2, 1, 1] [1, 5, 8, 9, 7, 5, 3, 2, 1, 1] [1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1] [1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1] [1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1] [1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1, 1] [1, 7, 19, 27, 30, 26, 21, 15, 11, 7, 5, 3, 2, 1, 1] [1, 8, 21, 34, 37, 35, 28, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 8, 24, 39, 47, 44, 38, 29, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 9, 27, 47, 57, 58, 49, 40, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 9, 30, 54, 70, 71, 65, 52, 41, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 10, 33, 64, 84, 90, 82, 70, 54, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 10, 37, 72, 101, 110, 105, 89, 73, 55, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 11, 40, 84, 119, 136, 131, 116, 94, 75, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 11, 44, 94, 141, 163, 164, 146, 123, 97, 76, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] [1, 12, 48, 108, 164, 199, 201, 186, 157, 128, 99, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1][1, 12, 52, 120, 192, 235, 248, 230, 201, 164, 131, 100, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
G(23) is 1255
0.043878 seconds (82.76 k allocations: 3.730 MiB)G(123) is 2552338241
0.064343 seconds (435.68 k allocations: 7.199 MiB)G(1234) is 156978797223733228787865722354959930
6.439370 seconds (43.57 M allocations: 723.421 MiB, 30.61% gc time)G(12345) is 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736 691.453611 seconds (4.32 G allocations: 71.973 GiB, 33.18% gc time)
Kotlin
<lang scala>import java.lang.Math.min import java.math.BigInteger import java.util.ArrayList import java.util.Arrays.asList
fun namesOfGod(n: Int): List<BigInteger> {
val cache = ArrayList<List<BigInteger>>() cache.add(asList(BigInteger.ONE))
(cache.size..n).forEach { l ->
val r = ArrayList<BigInteger>()
r.add(BigInteger.ZERO)
(1..l).forEach { x ->
r.add(r[r.size - 1] + cache[l - x][min(x, l - x)])
}
cache.add(r)
}
return cache[n]
}
fun row(n: Int) = namesOfGod(n).let { r -> (0 until n).map { r[it + 1] - r[it] } }
fun main(args: Array<String>) {
println("Rows:")
(1..25).forEach {
System.out.printf("%2d: %s%n", it, row(it))
}
println("\nSums:")
intArrayOf(23, 123, 1234, 1234).forEach {
val c = namesOfGod(it)
System.out.printf("%s %s%n", it, c[c.size - 1])
}
}</lang>
Rows: 1: [1] 2: [1, 1] 3: [1, 1, 1] 4: [1, 2, 1, 1] 5: [1, 2, 2, 1, 1] 6: [1, 3, 3, 2, 1, 1] 7: [1, 3, 4, 3, 2, 1, 1] 8: [1, 4, 5, 5, 3, 2, 1, 1] 9: [1, 4, 7, 6, 5, 3, 2, 1, 1] 10: [1, 5, 8, 9, 7, 5, 3, 2, 1, 1] 11: [1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1] 12: [1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1] 13: [1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1] 14: [1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1, 1] 15: [1, 7, 19, 27, 30, 26, 21, 15, 11, 7, 5, 3, 2, 1, 1] 16: [1, 8, 21, 34, 37, 35, 28, 22, 15, 11, 7, 5, 3, 2, 1, 1] 17: [1, 8, 24, 39, 47, 44, 38, 29, 22, 15, 11, 7, 5, 3, 2, 1, 1] 18: [1, 9, 27, 47, 57, 58, 49, 40, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 19: [1, 9, 30, 54, 70, 71, 65, 52, 41, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 20: [1, 10, 33, 64, 84, 90, 82, 70, 54, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 21: [1, 10, 37, 72, 101, 110, 105, 89, 73, 55, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 22: [1, 11, 40, 84, 119, 136, 131, 116, 94, 75, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 23: [1, 11, 44, 94, 141, 163, 164, 146, 123, 97, 76, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 24: [1, 12, 48, 108, 164, 199, 201, 186, 157, 128, 99, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] 25: [1, 12, 52, 120, 192, 235, 248, 230, 201, 164, 131, 100, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1] Sums: 23 1255 123 2552338241 1234 156978797223733228787865722354959930
Lasso
This code is derived from the Python solution, as an illustration of the difference in array behaviour (indexes, syntax), and loop and query expression as alternative syntax to "for". <lang Lasso>define cumu(n::integer) => { loop(-from=$cache->size,-to=#n+1) => { local(r = array(0), l = loop_count) loop(loop_count) => { protect => { #r->insert(#r->last + $cache->get(#l - loop_count)->get(math_min(loop_count+1, #l - loop_count))) } } #r->size > 1 ? $cache->insert(#r) } return $cache->get(#n) } define row(n::integer) => { // cache gets reset & rebuilt for each row, slower but more accurate var(cache = array(array(1))) local(r = cumu(#n+1)) local(o = array) loop(#n) => { protect => { #o->insert(#r->get(loop_count+1) - #r->get(loop_count)) } } return #o } 'rows:\r' loop(25) => {^ loop_count + ': '+ row(loop_count)->join(' ') + '\r' ^}
'sums:\r' with x in array(23, 123, 1234) do => {^ var(cache = array(array(1))) cumu(#x+1)->last '\r' ^}</lang>
- Output:
rows: 1: 1 2: 1 1 3: 1 1 1 4: 1 2 1 1 5: 1 2 2 1 1 6: 1 3 3 2 1 1 7: 1 3 4 3 2 1 1 8: 1 4 5 5 3 2 1 1 9: 1 4 7 6 5 3 2 1 1 10: 1 5 8 9 7 5 3 2 1 1 11: 1 5 10 11 10 7 5 3 2 1 1 12: 1 6 12 15 13 11 7 5 3 2 1 1 13: 1 6 14 18 18 14 11 7 5 3 2 1 1 14: 1 7 16 23 23 20 15 11 7 5 3 2 1 1 15: 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 16: 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 17: 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 18: 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 19: 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 20: 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 21: 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 22: 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 23: 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 24: 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 25: 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1 sums: 23: 1255 123: 2552338241 1234: 156978797223733228787865722354959930 12345: (ran long, timed out)
Lua
<lang lua>function nog(n)
local tri = Template:1 for r = 2, n do tri[r] = {} for c = 1, r do tri[r][c] = (tri[r-1][c-1] or 0) + (tri[r-c] and tri[r-c][c] or 0) end end return tri
end
function G(n)
local tri, sum = nog(n), 0 for _, v in ipairs(tri[n]) do sum = sum + v end return sum
end
tri = nog(25) for i, row in ipairs(tri) do
print(i .. ": " .. table.concat(row, " "))
end print("G(23) = " .. G(23)) print("G(123) = " .. G(123))</lang>
- Output:
1: 1 2: 1 1 3: 1 1 1 4: 1 2 1 1 5: 1 2 2 1 1 6: 1 3 3 2 1 1 7: 1 3 4 3 2 1 1 8: 1 4 5 5 3 2 1 1 9: 1 4 7 6 5 3 2 1 1 10: 1 5 8 9 7 5 3 2 1 1 11: 1 5 10 11 10 7 5 3 2 1 1 12: 1 6 12 15 13 11 7 5 3 2 1 1 13: 1 6 14 18 18 14 11 7 5 3 2 1 1 14: 1 7 16 23 23 20 15 11 7 5 3 2 1 1 15: 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 16: 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 17: 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 18: 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 19: 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 20: 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 21: 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 22: 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 23: 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 24: 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 25: 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1 G(23) = 1255 G(123) = 2552338241
Maple
<lang maple>TriangleLine(n) := map(rhs, Statistics :- Tally(map(x -> x[-1], combinat:-partition(n)))): Triangle := proc(m)
local i;
for i from 1 to m do
print(op(TriangleLine(i)));
end do
end proc:
</lang>
- Output:
Triangle(7);
1
1, 1
1, 1, 1
1, 2, 1, 1
1, 2, 2, 1, 1
1, 3, 3, 2, 1, 1
1, 3, 4, 3, 2, 1, 1
Mathematica / Wolfram Language
<lang mathematica>Table[Last /@ Reverse@Tally[First /@ IntegerPartitions[n]], {n, 10}] // Grid</lang>
- Output:
1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1
Here I use the bulit-in function PartitionsP to calculate . <lang mathematica>PartitionsP /@ {23, 123, 1234, 12345}</lang>
- Output:
{1255, 2552338241, 156978797223733228787865722354959930, 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736}
<lang mathematica>DiscretePlot[PartitionsP[n], {n, 1, 999}, PlotRange -> All]</lang>
Maxima
<lang maxima>for n thru 25 do print(makelist(length(integer_partitions(n-k,k)),k,1,n))$</lang>
- Output:
[1] [1,1] [1,1,1] [1,2,1,1] [1,2,2,1,1] [1,3,3,2,1,1] [1,3,4,3,2,1,1] [1,4,5,5,3,2,1,1] [1,4,7,6,5,3,2,1,1] [1,5,8,9,7,5,3,2,1,1] [1,5,10,11,10,7,5,3,2,1,1] [1,6,12,15,13,11,7,5,3,2,1,1] [1,6,14,18,18,14,11,7,5,3,2,1,1] [1,7,16,23,23,20,15,11,7,5,3,2,1,1] [1,7,19,27,30,26,21,15,11,7,5,3,2,1,1] [1,8,21,34,37,35,28,22,15,11,7,5,3,2,1,1] [1,8,24,39,47,44,38,29,22,15,11,7,5,3,2,1,1] [1,9,27,47,57,58,49,40,30,22,15,11,7,5,3,2,1,1] [1,9,30,54,70,71,65,52,41,30,22,15,11,7,5,3,2,1,1] [1,10,33,64,84,90,82,70,54,42,30,22,15,11,7,5,3,2,1,1] [1,10,37,72,101,110,105,89,73,55,42,30,22,15,11,7,5,3,2,1,1] [1,11,40,84,119,136,131,116,94,75,56,42,30,22,15,11,7,5,3,2,1,1] [1,11,44,94,141,163,164,146,123,97,76,56,42,30,22,15,11,7,5,3,2,1,1] [1,12,48,108,164,199,201,186,157,128,99,77,56,42,30,22,15,11,7,5,3,2,1,1] [1,12,52,120,192,235,248,230,201,164,131,100,77,56,42,30,22,15,11,7,5,3,2,1,1]
Using the built-in function to calculate : <lang maxima>makelist(num_partitions(n),n,[23,123,1234,12345]);</lang>
- Output:
(%o1) [1255, 2552338241, 156978797223733228787865722354959930, 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736]
Nim
<lang nim>import bigints
var cache = @[@[1.initBigInt]]
proc cumu(n: int): seq[BigInt] =
for m in cache.len .. n:
var r = @[0.initBigInt]
for x in 1..m:
r.add r[r.high] + cache[m-x][min(x, m-x)]
cache.add r
result = cache[n]
proc row(n: int): seq[BigInt] =
let r = cumu n result = @[] for i in 0 .. <n: result.add r[i+1] - r[i]
echo "rows:" for x in 1..10:
echo row x
echo "sums:" for x in [23, 123, 1234, 12345]:
let c = cumu(x) echo x, " ", c[c.high]</lang>
- Output:
@[1] @[1, 1] @[1, 1, 1] @[1, 2, 1, 1] @[1, 2, 2, 1, 1] @[1, 3, 3, 2, 1, 1] @[1, 3, 4, 3, 2, 1, 1] @[1, 4, 5, 5, 3, 2, 1, 1] @[1, 4, 7, 6, 5, 3, 2, 1, 1] @[1, 5, 8, 9, 7, 5, 3, 2, 1, 1] sums: 23 1255 123 2552338241 1234 156978797223733228787865722354959930 ^C
Faster version:
<lang nim>import bigints
var p = @[1.initBigInt]
proc partitions(n): BigInt =
p.add 0.initBigInt
for k in 1..n:
var d = n - k * (3 * k - 1) div 2
if d < 0:
break
if (k and 1) != 0:
p[n] += p[d]
else:
p[n] -= p[d]
d -= k
if d < 0:
break
if (k and 1) != 0:
p[n] += p[d]
else:
p[n] -= p[d]
result = p[p.high]
const ns = [23, 123, 1234, 12345] for i in 1 .. max(ns):
let p = partitions(i) if i in ns: echo i,": ",p</lang>
- Output:
23: 1255 123: 2552338241 1234: 156978797223733228787865722354959930 12345: 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
OCaml
<lang ocaml> let get, sum_unto =
let cache = ref [||]
let rec get i j =
if Array.length !cache < i then
cache :=
Array.init i begin fun i ->
try !cache.(i) with Invalid_argument _ ->
Array.make (i+1) (Num.num_of_int 0)
end;
if Num.(!cache.(i-1).(j-1) =/ num_of_int 0)
then !cache.(i-1).(j-1) <- sum_unto (i-j) j;
!cache.(i-1).(j-1)
and sum_unto i j =
let rec sum_unto sum i j =
match (i,j) with
|(0,0) -> (Num.num_of_int 1)
|(_,0) -> sum
|(i,j) when j > i -> sum_unto sum i i
|(i,j) -> sum_unto Num.(sum +/ (get i j)) i (j-1)
in
sum_unto (Num.num_of_int 0) i j
in
get, sum_unto
let sum_of_row n = sum_unto n n
let euler_recurrence =
let cache = ref [||] in
let rec recurrence = function
|n when n < 0 -> Num.num_of_int 0
|0 -> Num.num_of_int 1
|n ->
if n >= Array.length !cache then
cache :=
Array.init (n+1) (fun i ->
try !cache.(i) with Invalid_argument _ -> Num.num_of_int 0);
if Num.(!cache.(n) =/ num_of_int 0)
then begin
let rec summing sum = function
|0 -> sum
|k ->
let op = if k mod 2 = 0 then Num.sub_num else Num.add_num in
let sum = op sum (recurrence (n - k * (3*k - 1) / 2)) in
let sum = op sum (recurrence (n - k * (3*k + 1) / 2)) in
summing sum (k-1)
in
!cache.(n) <- summing (Num.num_of_int 0) n
end;
!cache.(n)
in
recurrence
let print i_max =
for i=1 to i_max do
print_int (i+1); print_string ": ";
for j=1 to i do
print_string (Num.string_of_num (get i j));
print_char ' ';
done;
print_newline ();
done
let () =
print 30;
print_newline ();
List.iter begin fun i ->
Printf.printf "%i: %s ?= %s\n" i
(Num.string_of_num (sum_of_row i))
(Num.string_of_num (euler_recurrence i));
flush stdout;
end
[23;123;1234;];
List.iter begin fun i ->
Printf.printf "%i: %s\n" i
(Num.string_of_num (euler_recurrence i));
flush stdout;
end
[23;123;1234;12345;123456]
</lang>
- Output:
2: 1 3: 1 1 4: 1 1 1 5: 1 2 1 1 6: 1 2 2 1 1 7: 1 3 3 2 1 1 8: 1 3 4 3 2 1 1 9: 1 4 5 5 3 2 1 1 10: 1 4 7 6 5 3 2 1 1 11: 1 5 8 9 7 5 3 2 1 1 12: 1 5 10 11 10 7 5 3 2 1 1 13: 1 6 12 15 13 11 7 5 3 2 1 1 14: 1 6 14 18 18 14 11 7 5 3 2 1 1 15: 1 7 16 23 23 20 15 11 7 5 3 2 1 1 16: 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 17: 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 18: 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 19: 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 20: 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 21: 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 22: 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 23: 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 24: 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 25: 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 26: 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1 27: 1 13 56 136 221 282 300 288 252 212 169 133 101 77 56 42 30 22 15 11 7 5 3 2 1 1 28: 1 13 61 150 255 331 364 352 318 267 219 172 134 101 77 56 42 30 22 15 11 7 5 3 2 1 1 29: 1 14 65 169 291 391 436 434 393 340 278 224 174 135 101 77 56 42 30 22 15 11 7 5 3 2 1 1 30: 1 14 70 185 333 454 522 525 488 423 355 285 227 175 135 101 77 56 42 30 22 15 11 7 5 3 2 1 1 31: 1 15 75 206 377 532 618 638 598 530 445 366 290 229 176 135 101 77 56 42 30 22 15 11 7 5 3 2 1 1 23: 1255 ?= 1255 123: 2552338241 ?= 2552338241 1234: 156978797223733228787865722354959930 ?= 156978797223733228787865722354959930 23: 1255 123: 2552338241 1234: 156978797223733228787865722354959930 12345: 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736 123456: 30817659578536496678545317146533980855296613274507139217608776782063054452191537379312358383342446230621170608408020911309259407611257151683372221925128388387168451943800027128045369650890220060901494540459081545445020808726917371699102825508039173543836338081612528477859613355349851184591540231790254269948278726548570660145691076819912972162262902150886818986555127204165221706149989 ./intnames 897.04s user 2.43s system 94% cpu 15:47.77 total
Ol
<lang scheme> (define (nine-billion-names row column)
(cond
((<= row 0) 0)
((<= column 0) 0)
((< row column) 0)
((= row 1) 1)
(else
(let ((addend (nine-billion-names (- row 1) (- column 1)))
(augend (nine-billion-names (- row column) column)))
(+ addend augend)))))
(define (print-row row)
(for-each (lambda (x)
(display (nine-billion-names row x))
(display " "))
(iota row 1)))
(define (print-triangle rows)
(for-each (lambda (x)
(print-row x)
(print))
(iota rows 1)))
(print-triangle 25) </lang>
- Output:
1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1 1 5 10 11 10 7 5 3 2 1 1 1 6 12 15 13 11 7 5 3 2 1 1 1 6 14 18 18 14 11 7 5 3 2 1 1 1 7 16 23 23 20 15 11 7 5 3 2 1 1 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
PARI/GP
<lang parigp>row(n)=my(v=vector(n)); forpart(i=n,v[i[#i]]++); v; show(n)=for(k=1,n,print(row(k))); show(25) apply(numbpart, [23,123,1234,12345]) plot(x=1,999.9, numbpart(x\1))</lang>
- Output:
[1]
[1, 1]
[1, 1, 1]
[1, 2, 1, 1]
[1, 2, 2, 1, 1]
[1, 3, 3, 2, 1, 1]
[1, 3, 4, 3, 2, 1, 1]
[1, 4, 5, 5, 3, 2, 1, 1]
[1, 4, 7, 6, 5, 3, 2, 1, 1]
[1, 5, 8, 9, 7, 5, 3, 2, 1, 1]
[1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1]
[1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1]
[1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1]
[1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1, 1]
[1, 7, 19, 27, 30, 26, 21, 15, 11, 7, 5, 3, 2, 1, 1]
[1, 8, 21, 34, 37, 35, 28, 22, 15, 11, 7, 5, 3, 2, 1, 1]
[1, 8, 24, 39, 47, 44, 38, 29, 22, 15, 11, 7, 5, 3, 2, 1, 1]
[1, 9, 27, 47, 57, 58, 49, 40, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
[1, 9, 30, 54, 70, 71, 65, 52, 41, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
[1, 10, 33, 64, 84, 90, 82, 70, 54, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
[1, 10, 37, 72, 101, 110, 105, 89, 73, 55, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
[1, 11, 40, 84, 119, 136, 131, 116, 94, 75, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2,1, 1]
[1, 11, 44, 94, 141, 163, 164, 146, 123, 97, 76, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
[1, 12, 48, 108, 164, 199, 201, 186, 157, 128, 99, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
[1, 12, 52, 120, 192, 235, 248, 230, 201, 164, 131, 100, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
%1 = [1255, 2552338241, 156978797223733228787865722354959930, 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736]
2.31e+031 |''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''"
| :
| :
| :
| :
| :|
| :|
| :|
| :|
| _|
| :|
| :|
| : |
| : |
| : |
| x |
| : |
| : |
| x |
| |
| _" |
1 ________________________________________________________xx,,,,,,
1 999.9
Using ploth in place of plot yields a nice image which cannot be uploaded at present.
Perl
<lang perl>use ntheory qw/:all/;
sub triangle_row {
my($n,@row) = (shift);
# Tally by first element of the unrestricted integer partitions.
forpart { $row[ $_[0] - 1 ]++ } $n;
@row;
}
printf "%2d: %s\n", $_, join(" ",triangle_row($_)) for 1..25; print "\n"; say "P($_) = ", partitions($_) for (23, 123, 1234, 12345);</lang>
- Output:
[rows are the same as below]
P(23) = 1255 P(123) = 2552338241 P(1234) = 156978797223733228787865722354959930 P(12345) = 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
<lang perl> use strict; use warnings;
- Where Raku uses arbitrary precision integers everywhere
- that you don't tell it not to do so, Perl 5 will only use
- them where you *do* tell it do so.
use Math::BigInt; use constant zero => Math::BigInt->bzero; use constant one => Math::BigInt->bone;
my @todo = [one]; my @sums = (zero); sub nextrow {
my $n = shift;
for my $l (@todo .. $n) {
$sums[$l] = zero;
#print "$l\r" if $l < $n;
my @r;
for my $x (reverse 0 .. $l-1) {
my $todo = $todo[$x];
$sums[$x] += shift @$todo if @$todo;
push @r, $sums[$x];
}
push @todo, \@r;
}
@{ $todo[$n] };
}
print "rows:\n"; for(1..25) {
printf("%2d: ", $_);
print join(' ', nextrow($_)), "\n";
} print "\nsums:\n"; for (23, 123, 1234, 12345) {
print $_, "." x (8 - length); my $i = 0; $i += $_ for nextrow($_); print $i, "\n";
} </lang>
- Output:
rows: 1: 1 2: 1 1 3: 1 1 1 4: 1 2 1 1 5: 1 2 2 1 1 6: 1 3 3 2 1 1 7: 1 3 4 3 2 1 1 8: 1 4 5 5 3 2 1 1 9: 1 4 7 6 5 3 2 1 1 10: 1 5 8 9 7 5 3 2 1 1 11: 1 5 10 11 10 7 5 3 2 1 1 12: 1 6 12 15 13 11 7 5 3 2 1 1 13: 1 6 14 18 18 14 11 7 5 3 2 1 1 14: 1 7 16 23 23 20 15 11 7 5 3 2 1 1 15: 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 16: 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 17: 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 18: 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 19: 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 20: 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 21: 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 22: 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 23: 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 24: 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 25: 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1 sums: 23......1243 123.....2552338241 1234....156978797223733228787865722354959930 ^C
Note: I didn't wait long enough to see what the next result was, and stopped the program.
Phix
-- demo\rosetta\9billionnames.exw with javascript_semantics sequence cache = {{1}} function cumu(integer n) sequence r for l=length(cache) to n do r = {0} for x=1 to l do r = append(r,r[-1]+cache[l-x+1][min(x,l-x)+1]) end for cache = append(cache,r) end for return cache[n] end function function row(integer n) sequence r = cumu(n+1) sequence res = repeat(0,n) for i=1 to n do res[i] = r[i+1]-r[i] end for return res end function for i=1 to 25 do puts(1,repeat(' ',50-2*i)) sequence r = row(i) for j=1 to i do printf(1,"%4d",r[j]) end for puts(1,"\n") end for
- Output:
1
1 1
1 1 1
1 2 1 1
1 2 2 1 1
1 3 3 2 1 1
1 3 4 3 2 1 1
1 4 5 5 3 2 1 1
1 4 7 6 5 3 2 1 1
1 5 8 9 7 5 3 2 1 1
1 5 10 11 10 7 5 3 2 1 1
1 6 12 15 13 11 7 5 3 2 1 1
1 6 14 18 18 14 11 7 5 3 2 1 1
1 7 16 23 23 20 15 11 7 5 3 2 1 1
1 7 19 27 30 26 21 15 11 7 5 3 2 1 1
1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1
1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1
1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1
1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1
1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1
1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1
1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1
1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1
1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1
1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
Part 2
with javascript_semantics include mpfr.e sequence p procedure calc(integer n) n += 1 for k=1 to n-1 do integer d = n - k * (3 * k - 1) / 2; if d<1 then exit end if if and_bits(k,1) then mpz_add(p[n],p[n],p[d]) else mpz_sub(p[n],p[n],p[d]) end if d -= k; if d<1 then exit end if if and_bits(k,1) then mpz_add(p[n],p[n],p[d]) else mpz_sub(p[n],p[n],p[d]) end if end for end procedure constant cx = {23, 123, 1234, 12345} puts(1,"sums:\n") integer at = 1 p = mpz_inits(cx[$]+1) mpz_set_si(p[1],1) for i=1 to cx[$] do calc(i) if i=cx[at] then printf(1,"%2d:%s\n",{i,mpz_get_str(p[i+1])}) at += 1 end if end for
- Output:
sums: 23:1255 123:2552338241 1234:156978797223733228787865722354959930 12345:69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
Third and last, a simple plot
include pGUI.e IupOpen() IupControlsOpen() Ihandle plot = IupPlot("MENUITEMPROPERTIES=Yes, SIZE=640x320") IupSetAttribute(plot, "TITLE", "9 Billion Names"); IupSetAttribute(plot, "TITLEFONTSIZE", "10"); IupSetAttribute(plot, "TITLEFONTSTYLE", "ITALIC"); IupSetAttribute(plot, "GRIDLINESTYLE", "DOTTED"); IupSetAttribute(plot, "GRID", "YES"); IupSetAttribute(plot, "AXS_XLABEL", "x"); IupSetAttribute(plot, "AXS_YLABEL", "G(x)"); IupSetAttribute(plot, "AXS_XFONTSTYLE", "ITALIC"); IupSetAttribute(plot, "AXS_YFONTSTYLE", "ITALIC"); IupSetAttribute(plot, "AXS_XSCALE", "LOG10"); IupSetAttribute(plot, "AXS_YSCALE", "LOG10"); IupSetAttribute(plot, "AXS_YTICKSIZEAUTO", "NO"); IupSetAttribute(plot, "AXS_YTICKMAJORSIZE", "8"); IupSetAttribute(plot, "AXS_YTICKMINORSIZE", "0"); IupPlotBegin(plot) for x=1 to 999 do IupPlotAdd(plot, x, sum(row(x))) -- (row() from part 1) end for {} = IupPlotEnd(plot) Ihandle dlg = IupDialog(plot) IupSetAttribute(dlg, "TITLE", "9 Billion Names") IupMap(dlg) IupShowXY(dlg,IUP_CENTER,IUP_CENTER) if platform()!=JS then IupMainLoop() IupClose() end if
PicoLisp
<lang PicoLisp>(de row (N)
(let C '((1))
(do N
(push 'C (grow C)) )
(mapcon
'((L)
(when (cdr L)
(cons (- (cadr L) (car L))) ) )
(car C) ) ) )
(de grow (Lst)
(let (L (length Lst) S 0)
(cons
0
(mapcar
'((I X)
(inc 'S
(get I (inc (min X (- L X)))) ) )
Lst
(range 1 L) ) ) ) )
(de sumr (N)
(let
(K 1
S 1
O (cons 1 (need N 0))
D
(make
(while
(<
(* K (dec (* 3 K)))
(* 2 N) )
(link (list (dec (* 2 K)) S))
(link (list K S))
(inc 'K)
(setq S (- S)) ) ) )
(for (Y O (cdr Y) (cdr Y))
(let Z Y
(for L D
(inc
(setq Z (cdr (nth Z (car L))))
(* (car Y) (cadr L)) ) ) ) )
(last O) ) )
(for I 25
(println (row I)) )
(bench
(for I '(23 123 1234 12345)
(println (sumr I)) ) )
(bye)</lang>
- Output:
(1) (1 1) (1 1 1) (1 2 1 1) (1 2 2 1 1) (1 3 3 2 1 1) (1 3 4 3 2 1 1) (1 4 5 5 3 2 1 1) (1 4 7 6 5 3 2 1 1) (1 5 8 9 7 5 3 2 1 1) (1 5 10 11 10 7 5 3 2 1 1) (1 6 12 15 13 11 7 5 3 2 1 1) (1 6 14 18 18 14 11 7 5 3 2 1 1) (1 7 16 23 23 20 15 11 7 5 3 2 1 1) (1 7 19 27 30 26 21 15 11 7 5 3 2 1 1) (1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1) (1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1) (1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1) (1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1) (1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1) (1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1) (1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1) (1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1) (1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1) (1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1) 1255 2552338241 156978797223733228787865722354959930 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736 0.626 sec
PureBasic
<lang purebasic> Define nMax.i=25, n.i, k.i Dim pfx.s(1)
Procedure.s Sigma(sx.s, sums.s)
Define i.i, v1.i, v2.i, r.i Define s.s, sa.s sums=ReverseString(sums) : s=ReverseString(sx) For i=1 To Len(s)*Bool(Len(s)>Len(sums))+Len(sums)*Bool(Len(sums)>=Len(s)) v1=Val(Mid(s,i,1)) v2=Val(Mid(sums,i,1)) r+v1+v2 sa+Str(r%10) r/10 Next i If r : sa+Str(r%10) : EndIf ProcedureReturn ReverseString(sa)
EndProcedure
Procedure.i Adr(row.i,col.i)
ProcedureReturn ((row-1)*row/2+col)*Bool(row>0 And col>0)
EndProcedure
Procedure Triangle(row.i,Array pfx.s(1))
Define n.i,k.i
Define zs.s
nMax=row
ReDim pfx(Adr(nMax,nMax))
For n=1 To nMax
For k=1 To n
If k>n : pfx(Adr(n,k))="0" : Continue : EndIf
If n=k : pfx(Adr(n,k))="1" : Continue : EndIf
If k<=n/2
zs=""
zs=Sigma(pfx(Adr(n-k,k)),zs)
zs=Sigma(pfx(Adr(n-1,k-1)),zs)
pfx(Adr(n,k))=zs
Else
pfx(Adr(n,k))=pfx(Adr(n-1,k-1))
EndIf
Next k
Next n
EndProcedure
Procedure.s sum(row.i, Array pfx.s(1))
Define s.s Triangle(row, pfx()) For n=1 To row s=Sigma(pfx(Adr(row,n)),s) Next n ProcedureReturn RSet(Str(row),5,Chr(32))+" : "+s
EndProcedure
OpenConsole()
Triangle(nMax, pfx()) For n=1 To nMax
Print(Space(((nMax*4-1)-(n*4-1))/2))
For k=1 To n
Print(RSet(pfx(Adr(n,k)),3,Chr(32))+Space(1))
Next k
PrintN("")
Next n PrintN("") PrintN(sum(23,pfx())) PrintN(sum(123,pfx())) PrintN(sum(1234,pfx())) PrintN(sum(12345,pfx())) Input() </lang>
- Output:
1
1 1
1 1 1
1 2 1 1
1 2 2 1 1
1 3 3 2 1 1
1 3 4 3 2 1 1
1 4 5 5 3 2 1 1
1 4 7 6 5 3 2 1 1
1 5 8 9 7 5 3 2 1 1
1 5 10 11 10 7 5 3 2 1 1
1 6 12 15 13 11 7 5 3 2 1 1
1 6 14 18 18 14 11 7 5 3 2 1 1
1 7 16 23 23 20 15 11 7 5 3 2 1 1
1 7 19 27 30 26 21 15 11 7 5 3 2 1 1
1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1
1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1
1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1
1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1
1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1
1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1
1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1
1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1
1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1
1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
23 : 1255
123 : 2552338241
1234 : 156978797223733228787865722354959930
12345 : 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
Python
<lang python>cache = 1 def cumu(n):
for l in range(len(cache), n+1):
r = [0]
for x in range(1, l+1):
r.append(r[-1] + cache[l-x][min(x, l-x)])
cache.append(r)
return cache[n]
def row(n):
r = cumu(n) return [r[i+1] - r[i] for i in range(n)]
print "rows:" for x in range(1, 11): print "%2d:"%x, row(x)
print "\nsums:"
for x in [23, 123, 1234, 12345]: print x, cumu(x)[-1]</lang>
- Output:
(I didn't actually wait long enough to see what the sum for 12345 is)
rows: 1: [1] 2: [1, 1] 3: [1, 1, 1] 4: [1, 2, 1, 1] 5: [1, 2, 2, 1, 1] 6: [1, 3, 3, 2, 1, 1] 7: [1, 3, 4, 3, 2, 1, 1] 8: [1, 4, 5, 5, 3, 2, 1, 1] 9: [1, 4, 7, 6, 5, 3, 2, 1, 1] 10: [1, 5, 8, 9, 7, 5, 3, 2, 1, 1] sums: 23 1255 123 2552338241 1234 156978797223733228787865722354959930 ^C
To calculate partition functions only: <lang python>def partitions(N):
diffs,k,s = [],1,1
while k * (3*k-1) < 2*N:
diffs.extend([(2*k - 1, s), (k, s)])
k,s = k+1,-s
out = [1] + [0]*N
for p in range(0, N+1):
x = out[p]
for (o,s) in diffs:
p += o
if p > N: break
out[p] += x*s
return out
p = partitions(12345) for x in [23,123,1234,12345]: print x, p[x]</lang>
This version uses only a fraction of the memory and of the running time, compared to the first one that has to generate all the rows:
<lang python>def partitions(n):
partitions.p.append(0)
for k in xrange(1, n + 1):
d = n - k * (3 * k - 1) // 2
if d < 0:
break
if k & 1:
partitions.p[n] += partitions.p[d]
else:
partitions.p[n] -= partitions.p[d]
d -= k
if d < 0:
break
if k & 1:
partitions.p[n] += partitions.p[d]
else:
partitions.p[n] -= partitions.p[d]
return partitions.p[-1]
partitions.p = [1]
def main():
ns = set([23, 123, 1234, 12345]) max_ns = max(ns)
for i in xrange(1, max_ns + 1):
if i > max_ns:
break
p = partitions(i)
if i in ns:
print "%6d: %s" % (i, p)
main()</lang>
- Output:
23: 1255 123: 2552338241 1234: 156978797223733228787865722354959930 12345: 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
Racket
<lang racket>#lang racket
(define (cdr-empty ls) (if (empty? ls) empty (cdr ls)))
(define (names-of n)
(define (names-of-tail ans raws-rest n)
(if (zero? n)
ans
(names-of-tail (cons 1 (append (map +
(take ans (length raws-rest))
(map car raws-rest))
(drop ans (length raws-rest))))
(filter (compose not empty?)
(map cdr-empty (cons ans raws-rest)))
(sub1 n))))
(names-of-tail '() '() n))
(define (G n) (foldl + 0 (names-of n)))
(module+ main
(build-list 25 (compose names-of add1)) (newline) (map G '(23 123 1234)))
</lang>
- Output:
'((1) (1 1) (1 1 1) (1 2 1 1) (1 2 2 1 1) (1 3 3 2 1 1) (1 3 4 3 2 1 1) (1 4 5 5 3 2 1 1) (1 4 7 6 5 3 2 1 1) (1 5 8 9 7 5 3 2 1 1) (1 5 10 11 10 7 5 3 2 1 1) (1 6 12 15 13 11 7 5 3 2 1 1) (1 6 14 18 18 14 11 7 5 3 2 1 1) (1 7 16 23 23 20 15 11 7 5 3 2 1 1) (1 7 19 27 30 26 21 15 11 7 5 3 2 1 1) (1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1) (1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1) (1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1) (1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1) (1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1) (1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1) (1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1) (1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1) (1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1) (1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1)) '(1255 2552338241 156978797223733228787865722354959930)
Raku
(formerly Perl 6) To save a bunch of memory, this algorithm throws away all the numbers that it knows it's not going to use again, on the assumption that the function will only be called with increasing values of $n. (It could easily be made to recalculate if it notices a regression.)
<lang perl6>my @todo = $[1]; my @sums = 0; sub nextrow($n) {
for +@todo .. $n -> $l {
my $r = [];
for reverse ^$l -> $x {
my @x := @todo[$x];
if @x {
$r.push: @sums[$x] += @x.shift;
}
else {
$r.push: @sums[$x];
}
}
@todo.push($r);
}
@todo[$n];
}
say "rows:"; say .fmt('%2d'), ": ", nextrow($_)[] for 1..25;
my @names-of-God = 1, { partition-sum ++$ } … *;
my @names-of-God-adder = lazy [\+] flat 1, ( (1 .. *) Z (1 .. *).map: * × 2 + 1 );
sub partition-sum ($n) {
sum @names-of-God[$n X- @names-of-God-adder[^(@names-of-God-adder.first: * > $n, :k)]]
Z× (flat (1, 1, -1, -1) xx *)
}
say "\nsums:"; for 23, 123, 1234, 12345 {
put $_, "\t", @names-of-God[$_];
}</lang>
- Output:
rows: 1: [1] 2: [1 1] 3: [1 1 1] 4: [1 2 1 1] 5: [1 2 2 1 1] 6: [1 3 3 2 1 1] 7: [1 3 4 3 2 1 1] 8: [1 4 5 5 3 2 1 1] 9: [1 4 7 6 5 3 2 1 1] 10: [1 5 8 9 7 5 3 2 1 1] 11: [1 5 10 11 10 7 5 3 2 1 1] 12: [1 6 12 15 13 11 7 5 3 2 1 1] 13: [1 6 14 18 18 14 11 7 5 3 2 1 1] 14: [1 7 16 23 23 20 15 11 7 5 3 2 1 1] 15: [1 7 19 27 30 26 21 15 11 7 5 3 2 1 1] 16: [1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1] 17: [1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1] 18: [1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1] 19: [1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1] 20: [1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1] 21: [1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1] 22: [1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1] 23: [1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1] 24: [1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1] 25: [1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1] sums: 23 1255 123 2552338241 1234 156978797223733228787865722354959930 12345 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
Red
<lang Rebol>Red [Purpose: "9 billion names of god with cache"]
cache: [ ;[rem start val] [1 1 1] [2 1 1] [2 2 1] ]
reset: [cache: [[1 1 1] [2 1 1] [2 2 1]]]
name-finder: function [max /every] [ repeat row max [ case [ row = 1 [if every [print "1"] continue] row = 2 [if every [prin "1 1"] continue] every [print ""] ] row-cache: make block! row - 1 repeat col row [ rem: start: val: 0 case [ col = row [ val: 1 if (not every) and (row = max) [print val + sum row-cache] ] not col = 1 [ rem: row - col start: min rem col val: cache-search rem start ] true val: 1 ] mem: reduce [row col val] append/only cache mem append row-cache val if every [prin [val ""]] ]]]
cache-search: function [rem start] [
val: 0
colvals: reduce []
foreach mem cache [
if (mem/1 = rem) and (mem/2 <= start) [
append colvals mem/3
]
]
val: sum colvals
return val
]
print "rows:" print name-finder/every 25 reduce reset print { sum:} name-finder 23 </lang>
- Output:
rows: 1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1 1 5 10 11 10 7 5 3 2 1 1 1 6 12 15 13 11 7 5 3 2 1 1 1 6 14 18 18 14 11 7 5 3 2 1 1 1 7 16 23 23 20 15 11 7 5 3 2 1 1 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1 sum: 1255
REXX
This REXX version displays a nicely "balanced" numbers triangle as per this task's requirement.
If the number of rows is entered as a signed positive integer, only the number of
partitions is shown,
(that is, the sum of the numbers on the last line of the number triangle).
If the number of rows is entered as a signed integer, the triangle isn't shown.
Memoization is used to quickly obtain information of previously calculated numbers
in the left-hand side of
the triangle and also previous calculated partitions.
The right half of the triangle isn't calculated but rather the value is taken from a previous row and column.
Also, the left two columns of the triangle are computed
directly [either 1 or row%2 (integer
divide)]
as well as the rightmost three
columns (either 1 or 2).
The formula used is:
which is derived from Euler's generating function. <lang rexx>/*REXX program generates and displays a number triangle for partitions of a number. */ numeric digits 400 /*be able to handle larger numbers. */ parse arg N . /*obtain optional argument from the CL.*/ if N== then N= 25 /*N specified? Then use the default. */ @.= 0; @.0= 1; aN= abs(N) /*initialize a partition number; AN abs*/ if N==N+0 then say ' G('aN"):" G(N) /*just do this for well formed numbers.*/
say 'partitions('aN"):" partitions(aN) /*do it the easy way.*/
exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ G: procedure; parse arg nn; !.= 0; !.4.2= 2; mx= 1; aN= abs(nn); build= nn>0
do j=1 for aN%2; !.j.j= 1 /*gen shortcuts for unity elements.*/
end /*j*/
do t=1 for 1+build; #.=1 /*generate triangle once or twice. */
do r=1 for aN; #.2= r % 2 /*#.2 is a shortcut calculation. */
do c=3 to r-2; #.c= gen#(r,c)
end /*c*/
L= length(mx); p= 0; __= /*__ will be a row of the triangle*/
do cc=1 for r; p= p + #.cc /*only sum the last row of numbers.*/
if \build then iterate /*should we skip building triangle?*/
mx= max(mx, #.cc) /*used to build the symmetric #s. */
__= __ right(#.cc, L) /*construct a row of the triangle. */
end /*cc*/
if t==1 then iterate /*Is this 1st time through? No show*/
say center( strip(__), 2 + (aN-1) * (length(mx) + 1) )
end /*r*/ /* [↑] center row of the triangle.*/
end /*t*/
return p /*return with the generated number.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/ gen#: procedure expose !.; parse arg x,y /*obtain the X and Y arguments.*/
if !.x.y\==0 then return !.x.y /*was number generated before ? */
if y>x%2 then do; nx= x+1-(y-x%2)*2-(x//2==0)
ny= nx % 2; !.x.y= !.nx.ny
return !.x.y /*return the calculated number. */
end /* [↑] right half of triangle. */
$= 1 /* [↓] left " " " */
do q=2 for y-1; xy= x-y; if q>xy then iterate
if q==2 then $= $ + xy % 2
else if q==xy-1 then $= $ + 1
else $= $ + gen#(xy,q) /*recurse.*/
end /*q*/
!.x.y=$; return $ /*use memoization; return with #.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/ partitions: procedure expose @.; parse arg n; if @.n\==0 then return @.n /* ◄─────┐*/
$= 0 /*Already known? Return ►────┘*/
do k=1 for n /*process N partitions. */
#= n - (k*3-1) * k % 2 /*calculate a partition number.*/
if #<0 then leave /*Is it negative? Then leave. */
if @.#==0 then x= partitions(#) /* [◄] this is a recursive call*/
else x= @.# /*the value is already known. */
#= # - k
if #<0 then y= 0 /*Is negative? Then use zero.*/
else if @.#==0 then y= partitions(p) /*recursive call.*/
else y= @.#
if k//2 then $= $ + x + y /*use this method if K is odd. */
else $= $ - x - y /* " " " " " " even.*/
end /*k*/ /* [↑] Euler's recursive func.*/
@.n= $; return $ /*use memoization; return num.*/</lang>
- output when using the default input (of 25 rows):
1
1 1
1 1 1
1 2 1 1
1 2 2 1 1
1 3 3 2 1 1
1 3 4 3 2 1 1
1 4 5 5 3 2 1 1
1 4 7 6 5 3 2 1 1
1 5 8 9 7 5 3 2 1 1
1 5 10 11 10 7 5 3 2 1 1
1 6 12 15 13 11 7 5 3 2 1 1
1 6 14 18 18 14 11 7 5 3 2 1 1
1 7 16 23 23 20 15 11 7 5 3 2 1 1
1 7 19 27 30 26 21 15 11 7 5 3 2 1 1
1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1
1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1
1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1
1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1
1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1
1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1
1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1
1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1
1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1
1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
G(25): 1958
partitions(25): 1958
- output when using the input: -23
G(23): 1255 partitions(23): 1255
- output when using the input: -123
G(123): 2552338241 partitions(123): 2552338241
- output when using the input: -1234
G(1234): 156978797223733228787865722354959930 partitions(1234): 156978797223733228787865722354959930
- output when using the input: -12345
G(12345): 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736 partitions(12345): 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
- output when using the input: +123456
partitions(123456): 30817659578536496678545317146533980855296613274507139217608776782063054452191537379312358383342446230621170608408020911309259407611257151683372221925128388387168451943800027128045369650890220060901494540459081545445020808726917371699102825508039173543836338081612528477859613355349851184591540231790254269948278726548570660145691076819912972162262902150886818986555127204165221706149989
(For the extra credit part) to view a horizontal histogram (plot) for the values for the number of partitions of 1 ──► 999 here at:
Ruby
Naive Solution
<lang ruby>
- Generate IPF triangle
- Nigel_Galloway: May 1st., 2013.
def g(n,g)
return 1 unless 1 < g and g < n-1
(2..g).inject(1){|res,q| res + (q > n-g ? 0 : g(n-g,q))}
end
(1..25).each {|n|
puts (1..n).map {|g| "%4s" % g(n,g)}.join
} </lang>
- Output:
1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1 1 5 10 11 10 7 5 3 2 1 1 1 6 12 15 13 11 7 5 3 2 1 1 1 6 14 18 18 14 11 7 5 3 2 1 1 1 7 16 23 23 20 15 11 7 5 3 2 1 1 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
Full Solution
<lang ruby>
- Find large values of IPF
- Nigel_Galloway: May 1st., 2013.
N = 12345 @ng = [] @ipn1 = [] @ipn2 = [] def g(n,g)
t = n-g-2 return 1 if n<4 or t<0 return @ng[g-2][n-4] unless n/2<g return @ipn1[t]
end @ng[0] = [] (4..N).each {|q| @ng[0][q-4] = 1 + g(q-2,2)} @ipn1[0] = @ng[0][0] @ipn2[0] = @ng[0][N-4] (1...(N/2-1)).each {|n|
@ng[n] = []
(n*2+4..N).each {|q| @ng[n][q-4] = g(q-1,n+1) + g(q-n-2,n+2)}
@ipn1[n] = @ng[n][n*2]
@ipn2[n] = @ng[n][N-4]
@ng[n-1] = nil
} @ipn2.pop if N.even?
puts "G(23) = #{@ipn1[21]}" puts "G(123) = #{@ipn1[121]}" puts "G(1234) = #{@ipn1[1232]}" n = 3 + @ipn1.inject(:+) + @ipn2.inject(:+) puts "G(12345) = #{n}" </lang>
- Output:
G(23) = 1255 G(123) = 2552338241 G(1234) = 156978797223733228787865722354959930 G(12345) = 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
Rust
<lang rust>extern crate num;
use std::cmp; use num::bigint::BigUint;
fn cumu(n: usize, cache: &mut Vec<Vec<BigUint>>) {
for l in cache.len()..n+1 {
let mut r = vec![BigUint::from(0u32)];
for x in 1..l+1 {
let prev = r[r.len() - 1].clone();
r.push(prev + cache[l-x][cmp::min(x, l-x)].clone());
}
cache.push(r);
}
}
fn row(n: usize, cache: &mut Vec<Vec<BigUint>>) -> Vec<BigUint> {
cumu(n, cache); let r = &cache[n]; let mut v: Vec<BigUint> = Vec::new();
for i in 0..n {
v.push(&r[i+1] - &r[i]);
}
v
}
fn main() {
let mut cache = vec![vec![BigUint::from(1u32)]];
println!("rows:");
for x in 1..26 {
let v: Vec<String> = row(x, &mut cache).iter().map(|e| e.to_string()).collect();
let s: String = v.join(" ");
println!("{}: {}", x, s);
}
println!("sums:");
for x in vec![23, 123, 1234, 12345] {
cumu(x, &mut cache);
let v = &cache[x];
let s = v[v.len() - 1].to_string();
println!("{}: {}", x, s);
}
}</lang>
- Output:
rows: 1: 1 2: 1 1 3: 1 1 1 4: 1 2 1 1 5: 1 2 2 1 1 6: 1 3 3 2 1 1 7: 1 3 4 3 2 1 1 8: 1 4 5 5 3 2 1 1 9: 1 4 7 6 5 3 2 1 1 10: 1 5 8 9 7 5 3 2 1 1 11: 1 5 10 11 10 7 5 3 2 1 1 12: 1 6 12 15 13 11 7 5 3 2 1 1 13: 1 6 14 18 18 14 11 7 5 3 2 1 1 14: 1 7 16 23 23 20 15 11 7 5 3 2 1 1 15: 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 16: 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 17: 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 18: 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 19: 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 20: 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 21: 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 22: 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 23: 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 24: 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 25: 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1 sums: 23: 1255 123: 2552338241 1234: 156978797223733228787865722354959930 12345: 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
Scala
Naive Solution
<lang scala> object Main {
// This is a special class for memoization
case class Memo[A,B](f: A => B) extends (A => B) {
private val cache = Map.empty[A, B] def apply(x: A) = cache getOrElseUpdate (x, f(x))
}
// Naive, but memoized solution
lazy val namesStartingMemo : Memo[Tuple2[Int, Int], BigInt] = Memo {
case (1, 1) => 1
case (a, n) =>
if (a > n/2) namesStartingMemo(a - 1, n - 1) else if (n < a) 0 else if (n == a) 1 else (1 to a).map(i => namesStartingMemo(i, n - a)).sum
}
def partitions(n: Int) = (1 to n).map(namesStartingMemo(_, n)).sum
// main method
def main(args: Array[String]): Unit = {
for (i <- 1 to 25) {
for (j <- 1 to i) {
print(namesStartingMemo(j, i)); print(' '); }
println() } println(partitions(23)) println(partitions(123)) println(partitions(1234)) println(partitions(12345)) }
} </lang>
- Output:
1
1 1
1 1 1
1 2 1 1
1 2 2 1 1
1 3 3 2 1 1
1 3 4 3 2 1 1
1 4 5 5 3 2 1 1
1 4 7 6 5 3 2 1 1
1 5 8 9 7 5 3 2 1 1
1 5 10 11 10 7 5 3 2 1 1
1 6 12 15 13 11 7 5 3 2 1 1
1 6 14 18 18 14 11 7 5 3 2 1 1
1 7 16 23 23 20 15 11 7 5 3 2 1 1
1 7 19 27 30 26 21 15 11 7 5 3 2 1 1
1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1
1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1
1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1
1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1
1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1
1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1
1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1
1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1
1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1
1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
1255
2552338241
156978797223733228787865722354959930
Exception in thread "main" java.lang.StackOverflowError
at scala.collection.mutable.HashTable$class.findEntry(HashTable.scala:130)
at scala.collection.mutable.HashMap.findEntry(HashMap.scala:39)
at scala.collection.mutable.HashMap.get(HashMap.scala:69)
at scala.collection.mutable.MapLike$class.getOrElseUpdate(MapLike.scala:187)
at scala.collection.mutable.AbstractMap.getOrElseUpdate(Map.scala:91)
at Main$Memo.apply(Main.scala:14)
...
(As you see, partitions(12345) fails with StackOverflowError)
Full Solution
<lang Scala>val cache = new Array[BigInt](15000) cache(0) = 1 val cacheNaive = scala.collection.mutable.Map[Tuple2[Int, Int], BigInt]()
def p(n: Int, k: Int): BigInt = cacheNaive.getOrElseUpdate((n, k), (n, k) match {
case (n, 1) => 1
case (n, k) if n < k => 0
case (n, k) if n == k => 1
case (n, k) =>
if (k > n/2) p(n - 1, k - 1)
else p(n - 1, k - 1) + p(n - k, k)
})
def partitions(n: Int) = (1 to n).map(p(n, _)).sum
def updateCache(n: Int, d: Int, k: Int) =
if ((k & 1) == 1) cache(n) = cache(n) + cache(d) else cache(n) = cache(n) - cache(d)
def quickPartitions(n: Int): BigInt = {
cache(n) = 0
for (k <- 1 to n) {
val d = n - k * (3 * k - 1) / 2
if (d >= 0) {
updateCache(n, d, k)
val e = d - k
if (e >= 0) {
updateCache(n, e, k)
}
}
}
cache(n)
}
for (i <- 1 to 23) {
for (j <- 1 to i) {
print(f"${p(i, j)}%4d")
}
println
} println(partitions(23))
for (i <- 1 until cache.length) {
quickPartitions(i)
} println(quickPartitions(123)) println(quickPartitions(1234)) println(quickPartitions(12345))</lang>
- Output:
1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 3 3 2 1 1 1 3 4 3 2 1 1 1 4 5 5 3 2 1 1 1 4 7 6 5 3 2 1 1 1 5 8 9 7 5 3 2 1 1 1 5 10 11 10 7 5 3 2 1 1 1 6 12 15 13 11 7 5 3 2 1 1 1 6 14 18 18 14 11 7 5 3 2 1 1 1 7 16 23 23 20 15 11 7 5 3 2 1 1 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 1255 2552338241 156978797223733228787865722354959930 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
scheme
<lang scheme>(define (f m n)
(define (sigma g x y)
(define (sum i)
(if (< i 0) 0 (+ (f x (- y i) ) (sum (- i 1)))))
(sum y))
(cond ((eq? m n) 1)
((eq? n 1) 1)
((eq? n 0) 0)
((< m n) (f m m))
((< (/ m 2) n) (sigma f (- m n) (- m n)))
(else (sigma f (- m n) n))))
(define (line m)
(define (connect i) (if (> i m) '() (cons (f m i) (connect (+ i 1))))) (connect 1))
(define (print x)
(define (print-loop i) (cond ((< i x) (begin (display (line i)) (display "\n") (print-loop (+ i 1)) )))) (print-loop 1))
(print 25)</lang>
- Output:
(1) (1 1) (1 1 1) (1 2 1 1) (1 2 2 1 1) (1 3 3 2 1 1) (1 3 4 3 2 1 1) (1 4 5 5 3 2 1 1) (1 4 7 6 5 3 2 1 1) (1 5 8 9 7 5 3 2 1 1) (1 5 10 11 10 7 5 3 2 1 1) (1 6 12 15 13 11 7 5 3 2 1 1) (1 6 14 18 18 14 11 7 5 3 2 1 1) (1 7 16 23 23 20 15 11 7 5 3 2 1 1) (1 7 19 27 30 26 21 15 11 7 5 3 2 1 1) (1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1) (1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1) (1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1) (1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1) (1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1) (1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1) (1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1) (1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1) (1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1)
Sidef
<lang ruby>var cache = 1
func cumu (n) {
for l (cache.len .. n) {
var r = [0]
for i (1..l) {
r << (r[-1] + cache[l-i][min(i, l-i)])
}
cache << r
}
cache[n]
}
func row (n) {
var r = cumu(n)
n.of {|i| r[i+1] - r[i] }
}
say "rows:" for i (1..15) {
"%2s: %s\n".printf(i, row(i))
}
say "\nsums:"
for i in [23, 123, 1234, 12345] {
"%2s : %4s\n".printf(i, cumu(i)[-1])
}</lang>
- Output:
rows: 1: [1] 2: [1, 1] 3: [1, 1, 1] 4: [1, 2, 1, 1] 5: [1, 2, 2, 1, 1] 6: [1, 3, 3, 2, 1, 1] 7: [1, 3, 4, 3, 2, 1, 1] 8: [1, 4, 5, 5, 3, 2, 1, 1] 9: [1, 4, 7, 6, 5, 3, 2, 1, 1] 10: [1, 5, 8, 9, 7, 5, 3, 2, 1, 1] 11: [1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1] 12: [1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1] 13: [1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1] 14: [1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1, 1] 15: [1, 7, 19, 27, 30, 26, 21, 15, 11, 7, 5, 3, 2, 1, 1] sums: 23 : 1255 123 : 2552338241 1234 : 156978797223733228787865722354959930 ^C
SPL
<lang spl>'print triangle > n, 1..25
k = 50-n*2
#.output(#.str("","<"+k+"<"),#.rs)
> k, 1..n
i = p(n,k)
s = #.str(i,">3<")
? k<n, s += " "+#.rs
#.output(s)
<
< p(n,k)=
? k=0 | k>n, <= 0 ? k=n, <= 1 <= p(n-1,k-1)+p(n-k,k)
.
'calculate partition function
- .output()
- .output("G(23) = ",g(23))
- .output("G(123) = ",g(123))
- .output("G(1234) = ",g(1234))
- .output("G(12345) = ",g(12345))
g(n)=
p[1] = 1
> i, 2..n+1
j = 2
k,p[i] = 0
> j>1
k += 1
j = i-#.lower((3*k*k+k)/2)
? j!<1, p[i] -= (-1)^k*p[j]
j = i-#.lower((3*k*k-k)/2)
? j!<1, p[i] -= (-1)^k*p[j]
<
<
<= p[n+1]
.</lang>
- Output:
1
1 1
1 1 1
1 2 1 1
1 2 2 1 1
1 3 3 2 1 1
1 3 4 3 2 1 1
1 4 5 5 3 2 1 1
1 4 7 6 5 3 2 1 1
1 5 8 9 7 5 3 2 1 1
1 5 10 11 10 7 5 3 2 1 1
1 6 12 15 13 11 7 5 3 2 1 1
1 6 14 18 18 14 11 7 5 3 2 1 1
1 7 16 23 23 20 15 11 7 5 3 2 1 1
1 7 19 27 30 26 21 15 11 7 5 3 2 1 1
1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1
1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1
1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1
1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1
1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1
1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1
1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1
1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1
1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1
1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1
G(23) = 1255
G(123) = 2552338241
G(1234) = 156978797223733228787865722354959930
G(12345) = 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
Stata
<lang stata>mata function part(n) { a = J(n,n,.) for (i=1;i<=n;i++) a[i,1] = a[i,i] = 1 for (i=3;i<=n;i++) { for (j=2;j<i;j++) a[i,j] = sum(a[i-j,1..min((j,i-j))]) } return(a) } end</lang>
The result is shown for n=10 to keep it small. Due to computations being done in floating point, the result is exact up to n=299, and suffers rounding for larger values of n. Compare the array with OEIS A008284 and row sums with OEIS A000041.
Output
: a = part(10)
: a
1 2 3 4 5 6 7 8 9 10
+---------------------------------------------------+
1 | 1 . . . . . . . . . |
2 | 1 1 . . . . . . . . |
3 | 1 1 1 . . . . . . . |
4 | 1 2 1 1 . . . . . . |
5 | 1 2 2 1 1 . . . . . |
6 | 1 3 3 2 1 1 . . . . |
7 | 1 3 4 3 2 1 1 . . . |
8 | 1 4 5 5 3 2 1 1 . . |
9 | 1 4 7 6 5 3 2 1 1 . |
10 | 1 5 8 9 7 5 3 2 1 1 |
+---------------------------------------------------+
: rowsum(a)'
1 2 3 4 5 6 7 8 9 10
+---------------------------------------------------+
1 | 1 2 3 5 7 11 15 22 30 42 |
+---------------------------------------------------+
Swift
<lang Swift>var cache = 1 func namesOfGod(n:Int) -> [Int] {
for l in cache.count...n {
var r = [0]
for x in 1...l {
r.append(r[r.count - 1] + cache[l - x][min(x, l-x)])
}
cache.append(r)
}
return cache[n]
}
func row(n:Int) -> [Int] {
let r = namesOfGod(n)
var returnArray = [Int]()
for i in 0...n - 1 {
returnArray.append(r[i + 1] - r[i])
}
return returnArray
}
println("rows:") for x in 1...25 {
println("\(x): \(row(x))")
}
println("\nsums: ")
for x in [23, 123, 1234, 12345] {
cache = 1 var array = namesOfGod(x) var numInt = array[array.count - 1] println("\(x): \(numInt)")
}</lang>
- Output:
rows:
1: [1]
2: [1, 1]
3: [1, 1, 1]
4: [1, 2, 1, 1]
5: [1, 2, 2, 1, 1]
6: [1, 3, 3, 2, 1, 1]
7: [1, 3, 4, 3, 2, 1, 1]
8: [1, 4, 5, 5, 3, 2, 1, 1]
9: [1, 4, 7, 6, 5, 3, 2, 1, 1]
10: [1, 5, 8, 9, 7, 5, 3, 2, 1, 1]
11: [1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1]
12: [1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1]
13: [1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1]
14: [1, 7, 16, 23, 23, 20, 15, 11, 7, 5, 3, 2, 1, 1]
15: [1, 7, 19, 27, 30, 26, 21, 15, 11, 7, 5, 3, 2, 1, 1]
16: [1, 8, 21, 34, 37, 35, 28, 22, 15, 11, 7, 5, 3, 2, 1, 1]
17: [1, 8, 24, 39, 47, 44, 38, 29, 22, 15, 11, 7, 5, 3, 2, 1, 1]
18: [1, 9, 27, 47, 57, 58, 49, 40, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
19: [1, 9, 30, 54, 70, 71, 65, 52, 41, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
20: [1, 10, 33, 64, 84, 90, 82, 70, 54, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
21: [1, 10, 37, 72, 101, 110, 105, 89, 73, 55, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
22: [1, 11, 40, 84, 119, 136, 131, 116, 94, 75, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
23: [1, 11, 44, 94, 141, 163, 164, 146, 123, 97, 76, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
24: [1, 12, 48, 108, 164, 199, 201, 186, 157, 128, 99, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
25: [1, 12, 52, 120, 192, 235, 248, 230, 201, 164, 131, 100, 77, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1]
sums:
23: 1255
123: 2552338241
1234: 156978797223733228787865722354959930
12345: 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
Tcl
<lang tcl>set cache 1 proc cumu {n} {
global cache
for {set l [llength $cache]} {$l <= $n} {incr l} {
set r 0 for {set x 1; set y [expr {$l-1}]} {$y >= 0} {incr x; incr y -1} { lappend r [expr { [lindex $r end] + [lindex $cache $y [expr {min($x, $y)}]] }] } lappend cache $r
} return [lindex $cache $n]
} proc row {n} {
set r [cumu $n]
for {set i 0; set j 1} {$j < [llength $r]} {incr i; incr j} {
lappend result [expr {[lindex $r $j] - [lindex $r $i]}]
} return $result
}
puts "rows:" foreach x {1 2 3 4 5 6 7 8 9 10} {
puts "${x}: \[[join [row $x] {, }]\]"
} puts "\nsums:" foreach x {23 123 1234 12345} {
puts "${x}: [lindex [cumu $x] end]"
}</lang>
- Output:
rows: 1: [1] 2: [1, 1] 3: [1, 1, 1] 4: [1, 2, 1, 1] 5: [1, 2, 2, 1, 1] 6: [1, 3, 3, 2, 1, 1] 7: [1, 3, 4, 3, 2, 1, 1] 8: [1, 4, 5, 5, 3, 2, 1, 1] 9: [1, 4, 7, 6, 5, 3, 2, 1, 1] 10: [1, 5, 8, 9, 7, 5, 3, 2, 1, 1] sums: 23: 1255 123: 2552338241 1234: 156978797223733228787865722354959930 ^C
(I killed the run when it started to take a significant proportion of my system's memory.)
VBA
<lang vb>Public Sub nine_billion_names()
Dim p(25, 25) As Long
p(1, 1) = 1
For i = 2 To 25
For j = 1 To i
p(i, j) = p(i - 1, j - 1) + p(i - j, j)
Next j
Next i
For i = 1 To 25
Debug.Print String$(50 - 2 * i, " ");
For j = 1 To i
Debug.Print String$(4 - Len(CStr(p(i, j))), " ") & p(i, j);
Next j
Debug.Print
Next i
End Sub</lang>
- Output:
1
1 1
1 1 1
1 2 1 1
1 2 2 1 1
1 3 3 2 1 1
1 3 4 3 2 1 1
1 4 5 5 3 2 1 1
1 4 7 6 5 3 2 1 1
1 5 8 9 7 5 3 2 1 1
1 5 10 11 10 7 5 3 2 1 1
1 6 12 15 13 11 7 5 3 2 1 1
1 6 14 18 18 14 11 7 5 3 2 1 1
1 7 16 23 23 20 15 11 7 5 3 2 1 1
1 7 19 27 30 26 21 15 11 7 5 3 2 1 1
1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1
1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1
1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1
1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1
Wren
<lang ecmascript>import "/big" for BigInt import "/fmt" for Fmt
var cache = BigInt.one var cumu = Fn.new { |n|
if (cache.count <= n) {
(cache.count..n).each { |l|
var r = [BigInt.zero]
(1..l).each { |x|
var min = l - x
if (x < min) min = x
r.add(r[-1] + cache[l - x][min])
}
cache.add(r)
}
}
return cache[n]
}
var row = Fn.new { |n|
var r = cumu.call(n)
return (0...n).map { |i| r[i+1] - r[i] }.toList
}
System.print("Rows:") (1..25).each { |i|
Fmt.print("$2d: $s", i, row.call(i))
}
System.print("\nSums:") [23, 123, 1234, 12345].each { |i|
Fmt.print("$5s: $s", i, cumu.call(i)[-1])
}</lang>
- Output:
Rows: 1: 1 2: 1 1 3: 1 1 1 4: 1 2 1 1 5: 1 2 2 1 1 6: 1 3 3 2 1 1 7: 1 3 4 3 2 1 1 8: 1 4 5 5 3 2 1 1 9: 1 4 7 6 5 3 2 1 1 10: 1 5 8 9 7 5 3 2 1 1 11: 1 5 10 11 10 7 5 3 2 1 1 12: 1 6 12 15 13 11 7 5 3 2 1 1 13: 1 6 14 18 18 14 11 7 5 3 2 1 1 14: 1 7 16 23 23 20 15 11 7 5 3 2 1 1 15: 1 7 19 27 30 26 21 15 11 7 5 3 2 1 1 16: 1 8 21 34 37 35 28 22 15 11 7 5 3 2 1 1 17: 1 8 24 39 47 44 38 29 22 15 11 7 5 3 2 1 1 18: 1 9 27 47 57 58 49 40 30 22 15 11 7 5 3 2 1 1 19: 1 9 30 54 70 71 65 52 41 30 22 15 11 7 5 3 2 1 1 20: 1 10 33 64 84 90 82 70 54 42 30 22 15 11 7 5 3 2 1 1 21: 1 10 37 72 101 110 105 89 73 55 42 30 22 15 11 7 5 3 2 1 1 22: 1 11 40 84 119 136 131 116 94 75 56 42 30 22 15 11 7 5 3 2 1 1 23: 1 11 44 94 141 163 164 146 123 97 76 56 42 30 22 15 11 7 5 3 2 1 1 24: 1 12 48 108 164 199 201 186 157 128 99 77 56 42 30 22 15 11 7 5 3 2 1 1 25: 1 12 52 120 192 235 248 230 201 164 131 100 77 56 42 30 22 15 11 7 5 3 2 1 1 Sums: 23: 1255 123: 2552338241 1234: 156978797223733228787865722354959930 12345: 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736
Yabasic
<lang Yabasic>clear screen
Sub nine_billion_names(rows)
local p(rows, rows), i, j, column
p(1, 1) = 1
For i = 2 To rows
For j = 1 To i
p(i, j) = p(i - 1, j - 1) + p(i - j, j)
Next j
Next i
For i = 1 To rows
column = rows * 2 - 2 * i - 2
For j = 1 To i
Print at(column + j * 4 + (1 - len(str$(p(i, j)))), i), p(i, j)
Next j
Next i
End Sub
nine_billion_names(20)</lang>
zkl
Takes its time getting to 100,000 but it does. Uses the GMP big int library. Does the big int math in place to avoid garbage creation. <lang zkl>var [const] BN=Import.lib("zklBigNum");
const N=0d100_000; p:=List.createLong(N+1,BN.fp(0),True); // (0,0,...) all different
fcn calc(n,p){
p[n].set(0); // reset array for each run
foreach k in ([1..n]){
d:=n - k *(3*k - 1)/2;
do(2){
if (d<0) break(2);
if (k.isOdd) p[n].add(p[d]); else p[n].sub(p[d]); d-=k;
} }
}</lang> <lang zkl>idx:=T(23, 123, 1234, 12345, 20000, 30000, 40000, 50000, N); p[0].set(1);
foreach i in (idx){
(1).pump(i,Void,calc.fp1(p)); // for n in [1..i] do calc(n,p) "%2d:\t%d".fmt(i,p[i]).println();
}</lang> The .fp/.fp1 methods create a closure, fixing the first or second parameter.
- Output:
23: 1255 123: 2552338241 1234: 156978797223733228787865722354959930 12345: 69420357953926116819562977205209384460667673094671463620270321700806074195845953959951425306140971942519870679768681736 ... 100000: 27493510569775696512677516320986352688173429315980054758203125984302147328114964173055050741660736621590157844774296248940493063070200461792764493033510116079342457190155718943509725312466108452006369558934464248716828789832182345009262853831404597021307130674510624419227311238999702284408609370935531629697851569569892196108480158600569421098519