Zeckendorf number representation
You are encouraged to solve this task according to the task description, using any language you may know.
Just as numbers can be represented in a positional notation as sums of multiples of the powers of ten (decimal) or two (binary); all the positive integers can be represented as the sum of one or zero times the distinct members of the Fibonacci series.
Recall that the first six distinct Fibonacci numbers are: 1, 2, 3, 5, 8, 13.
The decimal number eleven can be written as 0*13 + 1*8 + 0*5 + 1*3 + 0*2 + 0*1 or 010100 in positional notation where the columns represent multiplication by a particular member of the sequence. Leading zeroes are dropped so that 11 decimal becomes 10100.
10100 is not the only way to make 11 from the Fibonacci numbers however; 0*13 + 1*8 + 0*5 + 0*3 + 1*2 + 1*1 or 010011 would also represent decimal 11. For a true Zeckendorf number there is the added restriction that no two consecutive Fibonacci numbers can be used which leads to the former unique solution.
- Task
Generate and show here a table of the Zeckendorf number representations of the decimal numbers zero to twenty, in order.
The intention in this task to find the Zeckendorf form of an arbitrary integer. The Zeckendorf form can be iterated by some bit twiddling rather than calculating each value separately but leave that to another separate task.
- Also see
- OEIS A014417 for the the sequence of required results.
- Brown's Criterion - Numberphile
- Related task
11l
<lang 11l>V n = 20 F z(=n)
I n == 0
R [0]
V fib = [2, 1]
L fib[0] < n
fib = [sum(fib[0.<2])] [+] fib
[Int] dig
L(f) fib
I f <= n
dig [+]= 1
n -= f
E
dig [+]= 0
R I dig[0] {dig} E dig[1..]
L(i) 0..n
print(‘#3: #8’.format(i, z(i).map(d -> String(d)).join(‘’)))</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
360 Assembly
<lang 360asm>* Zeckendorf number representation 04/04/2017 ZECKEN CSECT
USING ZECKEN,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
STM R14,R12,12(R13) save previous context
ST R13,4(R15) link backward
ST R15,8(R13) link forward
LR R13,R15 set addressability
LA R6,0 i=0
DO WHILE=(C,R6,LE,=A(20)) do i=0 to 20
MVC PG,=CL80'xx : ' init buffer
LA R10,PG pgi=0
XDECO R6,XDEC i
MVC 0(2,R10),XDEC+10 output i
LA R10,5(R10) pgi+=5
MVC FIB,=A(1) fib(1)=1
MVC FIB+4,=A(2) fib(2)=2
LA R7,2 j=2
LR R1,R7 j
SLA R1,2 @fib(j)
DO WHILE=(C,R6,GT,FIB-4(R1) do while fib(j)<i
LA R7,1(R7) j++
LR R1,R7 j
SLA R1,2 ~
L R2,FIB-8(R1) fib(j-1)
A R2,FIB-12(R1) fib(j-2)
ST R2,FIB-4(R1) fib(j)=fib(j-1)+fib(j-2)
LR R1,R7 j
SLA R1,2 @fib(j)
ENDDO , enddo j
LR R8,R6 k=i
MVI BB,X'00' bb=false
DO WHILE=(C,R7,GE,=A(1)) do j=j to 1 by -1
LR R1,R7 j
SLA R1,2 ~
IF C,R8,GE,FIB-4(R1) THEN if fib(j)<=k then
MVI BB,X'01' bb=true
MVC 0(1,R10),=C'1' output '1'
LA R10,1(R10) pgi+=1
LR R1,R7 j
SLA R1,2 ~
S R8,FIB-4(R1) k=k-fib(j)
ELSE , else
IF CLI,BB,EQ,X'01' THEN if bb then
MVC 0(1,R10),=C'0' output '0'
LA R10,1(R10) pgi+=1
ENDIF , endif
ENDIF , endif
BCTR R7,0 j--
ENDDO , enddo j
IF CLI,BB,NE,X'01' THEN if not bb then
MVC 0(1,R10),=C'0' output '0'
ENDIF , endif
XPRNT PG,L'PG print buffer
LA R6,1(R6) i++
ENDDO , enddo i
L R13,4(0,R13) restore previous savearea pointer
LM R14,R12,12(R13) restore previous context
XR R15,R15 rc=0
BR R14 exit
FIB DS 32F Fibonnacci table BB DS X flag PG DS CL80 buffer XDEC DS CL12 temp
YREGS
END ZECKEN</lang>
- Output:
0 : 0 1 : 1 2 : 10 3 : 100 4 : 101 5 : 1000 6 : 1001 7 : 1010 8 : 10000 9 : 10001 10 : 10010 11 : 10100 12 : 10101 13 : 100000 14 : 100001 15 : 100010 16 : 100100 17 : 100101 18 : 101000 19 : 101001 20 : 101010
Action!
<lang Action!>PROC Encode(INT x CHAR ARRAY s)
INT ARRAY fib(22)= [1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657] INT i BYTE append
IF x=0 THEN s(0)=1 s(1)='0 RETURN FI
i=21 append=0
s(0)=0
WHILE i>=0
DO
IF x>=fib(i) THEN
x==-fib(i)
s(0)==+1
s(s(0))='1
append=1
ELSEIF append THEN
s(0)==+1
s(s(0))='0
FI
i==-1
OD
RETURN
PROC Main()
INT i CHAR ARRAY s(10)
FOR i=0 TO 20
DO
Encode(i,s)
PrintF("%I -> %S%E",i,s)
OD
RETURN</lang>
- Output:
Screenshot from Atari 8-bit computer
0 -> 0 1 -> 1 2 -> 10 3 -> 100 4 -> 101 5 -> 1000 6 -> 1001 7 -> 1010 8 -> 10000 9 -> 10001 10 -> 10010 11 -> 10100 12 -> 10101 13 -> 100000 14 -> 100001 15 -> 100010 16 -> 100100 17 -> 100101 18 -> 101000 19 -> 101001 20 -> 101010
Ada
<lang Ada>with Ada.Text_IO, Ada.Strings.Unbounded;
procedure Print_Zeck is
function Zeck_Increment(Z: String) return String is
begin
if Z="" then
return "1";
elsif Z(Z'Last) = '1' then
return Zeck_Increment(Z(Z'First .. Z'Last-1)) & '0';
elsif Z(Z'Last-1) = '0' then
return Z(Z'First .. Z'Last-1) & '1';
else -- Z has at least two digits and ends with "10"
return Zeck_Increment(Z(Z'First .. Z'Last-2)) & "00";
end if; end Zeck_Increment; use Ada.Strings.Unbounded; Current: Unbounded_String := Null_Unbounded_String;
begin
for I in 1 .. 20 loop
Current := To_Unbounded_String(Zeck_Increment(To_String(Current)));
Ada.Text_IO.Put(To_String(Current) & " ");
end loop;
end Print_Zeck;</lang>
- Output:
1 10 100 101 1000 1001 1010 10000 10001 10010 10100 10101 100000 100001 100010 100100 100101 101000 101001 101010
ALGOL 68
<lang algol68># print some Zeckendorf number representations #
- We handle 32-bit numbers, the maximum fibonacci number that can fit in a #
- 32 bit number is F(45) #
- build a table of 32-bit fibonacci numbers #
[ 45 ]INT fibonacci; fibonacci[ 1 ] := 1; fibonacci[ 2 ] := 2; FOR i FROM 3 TO UPB fibonacci DO fibonacci[ i ] := fibonacci[ i - 1 ] + fibonacci[ i - 2 ] OD;
- returns the Zeckendorf representation of n or "?" if one cannot be found #
PROC to zeckendorf = ( INT n )STRING:
IF n = 0 THEN
"0"
ELSE
STRING result := "";
INT f pos := UPB fibonacci;
INT rest := ABS n;
# find the first non-zero Zeckendorf digit #
WHILE f pos > LWB fibonacci AND rest < fibonacci[ f pos ] DO
f pos -:= 1
OD;
# if we found a digit, build the representation #
IF f pos >= LWB fibonacci THEN
# have a digit #
BOOL skip digit := FALSE;
WHILE f pos >= LWB fibonacci DO
IF rest <= 0 THEN
result +:= "0"
ELIF skip digit THEN
# we used the previous digit #
skip digit := FALSE;
result +:= "0"
ELIF rest < fibonacci[ f pos ] THEN
# can't use the digit at f pos #
skip digit := FALSE;
result +:= "0"
ELSE
# can use this digit #
skip digit := TRUE;
result +:= "1";
rest -:= fibonacci[ f pos ]
FI;
f pos -:= 1
OD
FI;
IF rest = 0 THEN
# found a representation #
result
ELSE
# can't find a representation #
"?"
FI
FI; # to zeckendorf #
FOR i FROM 0 TO 20 DO
print( ( whole( i, -3 ), " ", to zeckendorf( i ), newline ) )
OD </lang>
- Output:
0 0 1 1 2 10 3 100 4 101 5 1000 6 1001 7 1010 8 10000 9 10001 10 10010 11 10100 12 10101 13 100000 14 100001 15 100010 16 100100 17 100101 18 101000 19 101001 20 101010
AppleScript
(mapAccumuL example)
<lang AppleScript>--------------------- ZECKENDORF NUMBERS -------------------
-- zeckendorf :: Int -> String on zeckendorf(n)
script f
on |λ|(n, x)
if n < x then
[n, 0]
else
[n - x, 1]
end if
end |λ|
end script
if n = 0 then
{0} as string
else
item 2 of mapAccumL(f, n, |reverse|(just of tailMay(fibUntil(n)))) as string
end if
end zeckendorf
-- fibUntil :: Int -> [Int] on fibUntil(n)
set xs to {}
set limit to n
script atLimit
property ceiling : limit
on |λ|(x)
(item 2 of x) > (atLimit's ceiling)
end |λ|
end script
script nextPair
property series : xs
on |λ|([a, b])
set nextPair's series to nextPair's series & b
[b, a + b]
end |λ|
end script
|until|(atLimit, nextPair, {0, 1})
return nextPair's series
end fibUntil
TEST --------------------------
on run
intercalate(linefeed, ¬
map(zeckendorf, enumFromTo(0, 20)))
end run
GENERIC FUNCTIONS --------------------
-- enumFromTo :: Int -> Int -> [Int] on enumFromTo(m, n)
if m > n then
set d to -1
else
set d to 1
end if
set lst to {}
repeat with i from m to n by d
set end of lst to i
end repeat
return lst
end enumFromTo
-- foldl :: (a -> b -> a) -> a -> [b] -> a on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
-- 'The mapAccumL function behaves like a combination of map and foldl; -- it applies a function to each element of a list, passing an -- accumulating parameter from left to right, and returning a final -- value of this accumulator together with the new list.' (see Hoogle)
-- mapAccumL :: (acc -> x -> (acc, y)) -> acc -> [x] -> (acc, [y]) on mapAccumL(f, acc, xs)
script
on |λ|(a, x)
tell mReturn(f) to set pair to |λ|(item 1 of a, x)
[item 1 of pair, (item 2 of a) & item 2 of pair]
end |λ|
end script
foldl(result, [acc, []], xs)
end mapAccumL
-- map :: (a -> b) -> [a] -> [b] on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: Handler -> Script on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
-- intercalate :: Text -> [Text] -> Text on intercalate(strText, lstText)
set {dlm, my text item delimiters} to {my text item delimiters, strText}
set strJoined to lstText as text
set my text item delimiters to dlm
return strJoined
end intercalate
-- reverse :: [a] -> [a] on |reverse|(xs)
if class of xs is text then
(reverse of characters of xs) as text
else
reverse of xs
end if
end |reverse|
-- tailMay :: [a] -> Maybe [a] on tailMay(xs)
if length of xs > 1 then
{nothing:false, just:items 2 thru -1 of xs}
else
{nothing:true}
end if
end tailMay
-- until :: (a -> Bool) -> (a -> a) -> a -> a on |until|(p, f, x)
set mp to mReturn(p)
set v to x
tell mReturn(f)
repeat until mp's |λ|(v)
set v to |λ|(v)
end repeat
end tell
return v
end |until|</lang>
- Output:
0 1 10 100 101 1000 1001 1010 10000 10001 10010 10100 10101 100000 100001 100010 100100 100101 101000 101001 101010
Arturo
<lang rebol>Z: function [x][
if x=0 -> return "0"
fib: new [2 1]
n: new x
while -> n > first fib
-> insert 'fib 0 fib\0 + fib\1
result: new ""
loop fib 'f [
if? f =< n [
'result ++ "1"
'n - f
]
else -> 'result ++ "0"
]
if result\0 = `0` ->
result: slice result 1 (size result)-1
return result
]
loop 0..20 'i ->
print [pad to :string i 3 pad Z i 8]</lang>
- Output:
0 0 1 1 2 10 3 100 4 101 5 1000 6 1001 7 1010 8 10000 9 10001 10 10010 11 10100 12 10101 13 100000 14 100001 15 100010 16 100100 17 100101 18 101000 19 101001 20 101010
AutoHotkey
<lang AutoHotkey>Fib := NStepSequence(1, 2, 2, 20) Loop, 21 { i := A_Index - 1 , Out .= i ":`t", n := "" Loop, % Fib.MaxIndex() { x := Fib.MaxIndex() + 1 - A_Index if (Fib[x] <= i) n .= 1, i -= Fib[x] else n .= 0 } Out .= (n ? LTrim(n, "0") : 0) "`n" } MsgBox, % Out
NStepSequence(v1, v2, n, k) {
a := [v1, v2]
Loop, % k - 2 { a[j := A_Index + 2] := 0 Loop, % j < n + 2 ? j - 1 : n a[j] += a[j - A_Index] } return, a }</lang>NStepSequence() Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
AutoIt
<lang autoit> For $i = 0 To 20 ConsoleWrite($i &": "& Zeckendorf($i)&@CRLF) Next
Func Zeckendorf($int, $Fibarray = "") If Not IsArray($Fibarray) Then $Fibarray = Fibonacci($int) Local $ret = "" For $i = UBound($Fibarray) - 1 To 1 Step -1 If $Fibarray[$i] > $int And $ret = "" Then ContinueLoop ; dont use Leading Zeros If $Fibarray[$i] > $int Then $ret &= "0" Else If StringRight($ret, 1) <> "1" Then $ret &= "1" $int -= $Fibarray[$i] Else $ret &= "0" EndIf EndIf Next If $ret = "" Then $ret = "0" Return $ret EndFunc ;==>Zeckendorf
Func Fibonacci($max) $AList = ObjCreate("System.Collections.ArrayList") $AList.add("0") $current = 0 While True If $current > 1 Then $count = $AList.Count $current = $AList.Item($count - 1) $current = $current + $AList.Item($count - 2) Else $current += 1 EndIf $AList.add($current) If $current > $max Then ExitLoop WEnd $Array = $AList.ToArray Return $Array EndFunc ;==>Fibonacci </lang> Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
BBC BASIC
<lang bbcbasic> FOR n% = 0 TO 20
PRINT n% RIGHT$(" " + FNzeckendorf(n%), 8)
NEXT
PRINT '"Checking numbers up to 10000..."
FOR n% = 21 TO 10000
IF INSTR(FNzeckendorf(n%), "11") STOP
NEXT
PRINT "No Zeckendorf numbers contain consecutive 1's"
END
DEF FNzeckendorf(n%)
LOCAL i%, o$, fib%() : DIM fib%(45)
fib%(0) = 1 : fib%(1) = 1 : i% = 1
REPEAT
i% += 1
fib%(i%) = fib%(i%-1) + fib%(i%-2)
UNTIL fib%(i%) > n%
REPEAT
i% -= 1
IF n% >= fib%(i%) THEN
o$ += "1"
n% -= fib%(i%)
ELSE
o$ += "0"
ENDIF
UNTIL i% = 1
= o$</lang>
Output:
0 0
1 1
2 10
3 100
4 101
5 1000
6 1001
7 1010
8 10000
9 10001
10 10010
11 10100
12 10101
13 100000
14 100001
15 100010
16 100100
17 100101
18 101000
19 101001
20 101010
Checking numbers up to 10000...
No Zeckendorf numbers contain consecutive 1's
Befunge
The first number on the stack, 45*, specifies the range of values to display. However, the algorithm depends on a hardcoded list of Fibonacci values (currently just 10) so the theoretical maximum is 143. It's also constrained by the range of a Befunge data cell, which on many implementations will be as low as 127.
<lang befunge>45*83p0>:::.0`"0"v v53210p 39+!:,,9+< >858+37 *66g"7Y":v >3g`#@_^ v\g39$< ^8:+1,+5_5<>-:0\`| v:-\g39_^#:<*:p39< >0\`:!"0"+#^ ,#$_^</lang>
- Output:
0 0 1 1 2 10 3 100 4 101 5 1000 6 1001 7 1010 8 10000 9 10001 10 10010 11 10100 12 10101 13 100000 14 100001 15 100010 16 100100 17 100101 18 101000 19 101001 20 101010
C
<lang c>#include <stdio.h>
typedef unsigned long long u64;
- define FIB_INVALID (~(u64)0)
u64 fib[] = { 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073ULL, 4807526976ULL, 7778742049ULL, 12586269025ULL, 20365011074ULL, 32951280099ULL, 53316291173ULL, 86267571272ULL, 139583862445ULL, 225851433717ULL, 365435296162ULL, 591286729879ULL, 956722026041ULL, 1548008755920ULL, 2504730781961ULL, 4052739537881ULL, 6557470319842ULL, 10610209857723ULL, 17167680177565ULL,
27777890035288ULL // this 65-th one is for range check };
u64 fibbinary(u64 n) { if (n >= fib[64]) return FIB_INVALID;
u64 ret = 0; int i; for (i = 64; i--; ) if (n >= fib[i]) { ret |= 1ULL << i; n -= fib[i]; }
return ret; }
void bprint(u64 n, int width) { if (width > 64) width = 64;
u64 b; for (b = 1ULL << (width - 1); b; b >>= 1) putchar(b == 1 && !n ? '0' : b > n ? ' ' : b & n ? '1' : '0'); putchar('\n'); }
int main(void) { int i;
for (i = 0; i <= 20; i++) printf("%2d:", i), bprint(fibbinary(i), 8);
return 0; }</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
C#
<lang csharp> using System; using System.Collections.Generic; using System.Linq; using System.Text;
namespace Zeckendorf {
class Program
{
private static uint Fibonacci(uint n)
{
if (n < 2)
{
return n;
}
else
{
return Fibonacci(n - 1) + Fibonacci(n - 2);
}
}
private static string Zeckendorf(uint num)
{
IList<uint> fibonacciNumbers = new List<uint>();
uint fibPosition = 2;
uint currentFibonaciNum = Fibonacci(fibPosition);
do
{
fibonacciNumbers.Add(currentFibonaciNum);
currentFibonaciNum = Fibonacci(++fibPosition);
} while (currentFibonaciNum <= num);
uint temp = num;
StringBuilder output = new StringBuilder();
foreach (uint item in fibonacciNumbers.Reverse())
{
if (item <= temp)
{
output.Append("1");
temp -= item;
}
else
{
output.Append("0");
}
}
return output.ToString();
}
static void Main(string[] args)
{
for (uint i = 1; i <= 20; i++)
{
string zeckendorfRepresentation = Zeckendorf(i);
Console.WriteLine(string.Format("{0} : {1}", i, zeckendorfRepresentation));
}
Console.ReadKey();
}
}
} </lang>
- Output:
1 : 1 2 : 10 3 : 100 4 : 101 5 : 1000 6 : 1001 7 : 1010 8 : 10000 9 : 10001 10 : 10010 11 : 10100 12 : 10101 13 : 100000 14 : 100001 15 : 100010 16 : 100100 17 : 100101 18 : 101000 19 : 101001 20 : 101010
C++
Using a C++11 User Defined Literal
see Here for a further example using this class. <lang cpp> // For a class N which implements Zeckendorf numbers: // I define an increment operation ++() // I define a comparison operation <=(other N) // Nigel Galloway October 22nd., 2012
- include <iostream>
class N { private:
int dVal = 0, dLen;
public:
N(char const* x = "0"){
int i = 0, q = 1;
for (; x[i] > 0; i++);
for (dLen = --i/2; i >= 0; i--) {
dVal+=(x[i]-48)*q;
q*=2;
}}
const N& operator++() {
for (int i = 0;;i++) {
if (dLen < i) dLen = i;
switch ((dVal >> (i*2)) & 3) {
case 0: dVal += (1 << (i*2)); return *this;
case 1: dVal += (1 << (i*2)); if (((dVal >> ((i+1)*2)) & 1) != 1) return *this;
case 2: dVal &= ~(3 << (i*2));
}}}
const bool operator<=(const N& other) const {return dVal <= other.dVal;}
friend std::ostream& operator<<(std::ostream&, const N&);
}; N operator "" N(char const* x) {return N(x);} std::ostream &operator<<(std::ostream &os, const N &G) {
const static std::string dig[] {"00","01","10"}, dig1[] {"","1","10"};
if (G.dVal == 0) return os << "0";
os << dig1[(G.dVal >> (G.dLen*2)) & 3];
for (int i = G.dLen-1; i >= 0; i--) os << dig[(G.dVal >> (i*2)) & 3];
return os;
} </lang> I may now write: <lang cpp> int main(void) { //for (N G; G <= 101010N; ++G) std::cout << G << std::endl; // from zero to 101010M
for (N G(101N); G <= 101010N; ++G) std::cout << G << std::endl; // from 101N to 101010N return 0;
} </lang> Which produces:
- Output:
101 1000 1001 1010 10000 10001 10010 10100 10101 100000 100001 100010 100100 100101 101000 101001 101010
Clojure
<lang clojure>(def fibs (lazy-cat [1 1] (map + fibs (rest fibs))))
(defn z [n]
(if (zero? n)
"0"
(let [ps (->> fibs (take-while #(<= % n)) rest reverse)
fz (fn [[s n] p]
(if (>= n p)
[(conj s 1) (- n p)]
[(conj s 0) n]))]
(->> ps (reduce fz [[] n]) first (apply str)))))
(doseq [n (range 0 21)] (println n (z n)))</lang>
CLU
<lang clu>% Get list of distinct Fibonacci numbers up to N fibonacci = proc (n: int) returns (array[int])
list: array[int] := array[int]$[]
a: int := 1
b: int := 2
while a <= n do
array[int]$addh(list,a)
a, b := b, a+b
end
return(list)
end fibonacci
% Find the Zeckendorf representation of N zeckendorf = proc (n: int) returns (string) signals (negative)
if n<0 then signal negative end
if n=0 then return("0") end
fibs: array[int] := fibonacci(n)
result: array[char] := array[char]$[]
while ~array[int]$empty(fibs) do
fib: int := array[int]$remh(fibs)
if fib <= n then
n := n - fib
array[char]$addh(result,'1')
else
array[char]$addh(result,'0')
end
end
return(string$ac2s(result))
end zeckendorf
% Print the Zeckendorf representations of 0 to 20 start_up = proc ()
po: stream := stream$primary_output()
for i: int in int$from_to(0,20) do
stream$putright(po, int$unparse(i), 2)
stream$puts(po, ": ")
stream$putl(po, zeckendorf(i))
end
end start_up</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Common Lisp
Common Lisp's arbitrary precision integers should handle any positive input: <lang lisp>(defun zeckendorf (n)
"returns zeckendorf integer of n (see OEIS A003714)" (let ((fib '(2 1)))
;; extend Fibonacci sequence long enough (loop while (<= (car fib) n) do (push (+ (car fib) (cadr fib)) fib)) (loop with r = 0 for f in fib do (setf r (* 2 r)) (when (>= n f) (setf n (- n f)) (incf r)) finally (return r))))
- task requirement
(loop for i from 0 to 20 do
(format t "~2D: ~2R~%" i (zeckendorf i)))</lang>
<lang lisp>
- Print Zeckendorf numbers upto 20.
- I have implemented this as a state machine.
- Nigel Galloway - October 13th., 2012
(let ((fibz '(13 8 5 3 2 1))) (dotimes (G 21) (progn (format t "~S is " G)
(let ((z 0) (ng G)) (dolist (N fibz)
(if (> z 1) (progn (setq z 1) (format t "~S" 0))
(if (>= ng N) (progn (setq z 2) (setq ng (- ng N)) (format t "~S" 1))
(if (= z 1) (format t "~S" 0)))))
(if (= z 0) (format t "~S~%" 0) (format t "~%"))))))
</lang>
- Output:
0 is 0 1 is 1 2 is 10 3 is 100 4 is 101 5 is 1000 6 is 1001 7 is 1010 8 is 10000 9 is 10001 10 is 10010 11 is 10100 12 is 10101 13 is 100000 14 is 100001 15 is 100010 16 is 100100 17 is 100101 18 is 101000 19 is 101001 20 is 101010
Cowgol
<lang cowgol>include "cowgol.coh";
sub zeckendorf(n: uint32, buf: [uint8]): (r: [uint8]) is
var fibs: uint32[] := {
0,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,
2584,4181,6765,10946,17711,28657,46368,75025,121393,
196418,317811,514229,832040,1346269,2178309,3524578,
5702887,9227465,14930352,24157817,39088169,63245986,
102334155,165580141,267914296,433494437,701408733,
1134903170,1836311903,2971215073
};
r := buf;
if n == 0 then
[r] := '0';
[@next r] := 0;
return;
end if;
var fib: [uint32] := &fibs[1];
while n >= [fib] loop
fib := @next fib;
end loop;
fib := @prev fib;
while [fib] != 0 loop
if [fib] <= n then
n := n - [fib];
[buf] := '1';
else
[buf] := '0';
end if;
fib := @prev fib;
buf := @next buf;
end loop;
[buf] := 0;
end sub;
var i: uint32 := 0; while i <= 20 loop
print_i32(i);
print(": ");
print(zeckendorf(i, LOMEM));
print_nl();
i := i + 1;
end loop;</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Crystal
<lang ruby>def zeckendorf(n)
return 0 if n.zero?
fib = [1, 2]
while fib[-1] < n; fib << fib[-2] + fib[-1] end
digit = ""
fib.reverse_each do |f|
if f <= n
digit, n = digit + "1", n - f
else
digit += "0"
end
end
digit.to_i
end
(0..20).each { |i| puts "%3d: %8d" % [i, zeckendorf(i)] }</lang>
Using an explicit Iterator. <lang ruby>class ZeckendorfIterator
include Iterator(String)
def initialize @x = 0 end
def next
bin = @x.to_s(2)
@x += 1
while bin.includes?("11")
bin = @x.to_s(2)
@x += 1
end
bin
end
end
def zeckendorf(n)
ZeckendorfIterator.new.first(n)
end
zeckendorf(21).each_with_index{ |x,i| puts "%3d: %8s"% [i, x] }</lang>
Using oneliners. <lang ruby>def zeckendorf(n)
0.step.map(&.to_s(2)).reject(&.includes?("11")).first(n)
end
- or a little faster
def zeckendorf(n)
0.step.compact_map{ |x| bin = x.to_s(2); bin unless bin.includes?("11") }.first(n)
end
zeckendorf(21).each_with_index{ |x,i| puts "%3d: %8s"% [i, x] }</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
D
<lang d>import std.stdio, std.range, std.algorithm, std.functional;
void main() {
size_t
.max
.iota
.filter!q{ !(a & (a >> 1)) }
.take(21)
.binaryReverseArgs!writefln("%(%b\n%)");
}</lang>
- Output:
0 1 10 100 101 1000 1001 1010 10000 10001 10010 10100 10101 100000 100001 100010 100100 100101 101000 101001 101010
<lang d>import std.stdio, std.typecons;
int zeckendorf(in int n) pure nothrow {
Tuple!(int,"remaining", int,"set")
zr(in int fib0, in int fib1, in int n, in uint bit) pure nothrow {
if (fib1 > n)
return typeof(return)(n, 0);
auto rs = zr(fib1, fib0 + fib1, n, bit + 1);
if (fib1 <= rs.remaining) {
rs.set |= 1 << bit;
rs.remaining -= fib1;
}
return rs;
}
return zr(1, 1, n, 0)[1];
}
void main() {
foreach (i; 0 .. 21)
writefln("%2d: %6b", i, zeckendorf(i));
}</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
(Same output.) <lang d>import std.stdio, std.algorithm, std.range;
string zeckendorf(size_t n) {
if (n == 0)
return "0";
auto fibs = recurrence!q{a[n - 1] + a[n - 2]}(1, 2);
string result;
foreach_reverse (immutable f; fibs.until!(x => x > n).array) {
result ~= (f <= n) ? '1' : '0';
if (f <= n)
n -= f;
}
return result;
}
void main() {
foreach (immutable i; 0 .. 21)
writefln("%2d: %6s", i, i.zeckendorf);
}</lang>
EchoLisp
We analytically find the first fibonacci(i) >= n, using the formula i = log((n* Φ) + 0.5) / log(Φ) . <lang scheme>
- special fib's starting with 1 2 3 5 ...
(define (fibonacci n)
(+ (fibonacci (1- n)) (fibonacci (- n 2))))
(remember 'fibonacci #(1 2))
(define-constant Φ (// (1+ (sqrt 5)) 2)) (define-constant logΦ (log Φ))
- find i
- fib(i) >= n
(define (iFib n)
(floor (// (log (+ (* n Φ) 0.5)) logΦ)))
- left trim zeroes
(string-delimiter "") (define (zeck->string digits)
(if (!= 0 (first digits))
(string-join digits "")
(zeck->string (rest digits))))
(define (Zeck n)
(cond
(( < n 0) "no negative zeck")
((inexact? n) "no floating zeck")
((zero? n) "0")
(else (zeck->string
(for/list ((s (reverse (take fibonacci (iFib n)))))
(if ( > s n) 0
(begin (-= n s) 1 )))))))
</lang>
- Output:
(take Zeck 21)
→ (0 1 10 100 101 1000 1001 1010 10000 10001
10010 10100 10101 100000 100001 100010 100100 100101 101000 101001 101010)
(Zeck 1000000000)
→ 1010000100100001010101000001000101000101001
Elena
ELENA 4.x : <lang elena>import system'routines; import system'collections; import system'text; import extensions; extension op {
fibonacci()
{
if (self < 2)
{
^ self
}
else
{
^ (self - 1).fibonacci() + (self - 2).fibonacci()
};
}
zeckendorf()
{
var fibonacciNumbers := new List<int>();
int num := self;
int fibPosition := 2;
int currentFibonaciNum := fibPosition.fibonacci();
while (currentFibonaciNum <= num)
{
fibonacciNumbers.append:currentFibonaciNum;
fibPosition := fibPosition + 1;
currentFibonaciNum := fibPosition.fibonacci()
};
auto output := new TextBuilder();
int temp := num;
fibonacciNumbers.sequenceReverse().forEach:(item)
{
if (item <= temp)
{
output.write("1");
temp := temp - item
}
else
{
output.write("0")
}
};
^ output.Value
}
}
public program() {
for(int i := 1, i <= 20, i += 1)
{
console.printFormatted("{0} : {1}",i,i.zeckendorf()).writeLine()
};
console.readChar()
}</lang>
- Output:
1 : 1 2 : 10 3 : 100 4 : 101 5 : 1000 6 : 1001 7 : 1010 8 : 10000 9 : 10001 10 : 10010 11 : 10100 12 : 10101 13 : 100000 14 : 100001 15 : 100010 16 : 100100 17 : 100101 18 : 101000 19 : 101001 20 : 101010
Elixir
Stream generator: <lang elixir>defmodule Zeckendorf do
def number do
Stream.unfold(0, fn n -> zn_loop(n) end)
end
defp zn_loop(n) do
bin = Integer.to_string(n, 2)
if String.match?(bin, ~r/11/), do: zn_loop(n+1), else: {bin, n+1}
end
end
Zeckendorf.number |> Enum.take(21) |> Enum.with_index |> Enum.each(fn {zn, i} -> IO.puts "#{i}: #{zn}" end)</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Fibonacci numbers: <lang elixir>defmodule Zeckendorf do
def number(n) do
fib_loop(n, [2,1])
|> Enum.reduce({"",n}, fn f,{dig,i} ->
if f <= i, do: {dig<>"1", i-f}, else: {dig<>"0", i}
end)
|> elem(0) |> String.to_integer
end
defp fib_loop(n, fib) when n < hd(fib), do: fib
defp fib_loop(n, [a,b|_]=fib), do: fib_loop(n, [a+b | fib])
end
for i <- 0..20, do: IO.puts "#{i}: #{Zeckendorf.number(i)}"</lang> same output
F#
<lang fsharp>let fib = Seq.unfold (fun (x, y) -> Some(x, (y, x + y))) (1,2)
let zeckendorf n =
if n = 0 then ["0"]
else
let folder k state =
let (n, z) = (fst state), (snd state)
if n >= k then (n - k, "1" :: z)
else (n, "0" :: z)
let fb = fib |> Seq.takeWhile (fun i -> i<=n) |> Seq.toList
snd (List.foldBack folder fb (n, []))
|> List.rev
for i in 0 .. 20 do printfn "%2d: %8s" i (String.concat "" (zeckendorf i))</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Factor
<lang factor>USING: formatting kernel locals make math sequences ;
- fib<= ( n -- seq )
1 2 [ [ dup n <= ] [ 2dup + [ , ] 2dip ] while drop , ]
{ } make ;
- zeck ( n -- str )
0 :> s! n fib<= <reversed> [ dup s + n <= [ s + s! 49 ] [ drop 48 ] if ] "" map-as ;
21 <iota> [ dup zeck "%2d: %6s\n" printf ] each</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Forth
<lang forth>: fib<= ( n -- n )
>r 0 1 BEGIN dup r@ <= WHILE tuck + REPEAT drop rdrop ;
- z. ( n -- )
dup fib<= dup . -
BEGIN ?dup WHILE
dup fib<= dup [char] + emit space . -
REPEAT ;
- tab 9 emit ;
- zeckendorf ( -- )
21 0 DO
cr i 2 .r tab i z.
LOOP ;</lang>
- Output:
zeckendorf 0 0 1 1 2 2 3 3 4 3 + 1 5 5 6 5 + 1 7 5 + 2 8 8 9 8 + 1 10 8 + 2 11 8 + 3 12 8 + 3 + 1 13 13 14 13 + 1 15 13 + 2 16 13 + 3 17 13 + 3 + 1 18 13 + 5 19 13 + 5 + 1 20 13 + 5 + 2 ok
Fortran
The simplest representation of a number in the Zeckendorf manner is as a sequence of digits, such as are used in multi-precision arithmetic, and for this an array of integers will do. Rather excessively, as only two states are required and the default integer style is usually thirty-two bits these days. Some compilers allow the specification of one-byte integers, as in INTEGER*1 D(0:ZLAST) so that would be only an eight-fold excess. One could escalate to fiddling with individual bits within a number (as is done in Extensible_prime_generator#Fortran) and a 16-bit integer would be adequate for the specified tests, but Fortran syntax has not been extended to offer simple methods for manipulating bits such as D(7:7) to obtain the seventh bit of D. Instead one might use special library routines as supplied by F90 such as IBITS(D,7,1) for the same effect, though possibly at a cost in code size and execution time. Less storage may be saved through cramming bits than is consumed by the code needed to extract individual bits. Such values could then be displayed using theB format code. However, the source code would now be littered with the details of bit access rather than the form of the Zeckendorf procedure.
An alternative lies in noting that only the sequences 00, 01, and 10 can appear (because 11 is unnecessary; see below), so a base three scheme could be used to represent the three such pairs of bits. But this still contains redundancies. Suppose a 01 value is somewhere in the sequence: then it may be followed only by 00 or 01, and likewise 10 be preceded only by 10 or 00; just two values, not three. Perhaps a still more cunning compaction scheme could be devised to take advantage of these details, or some other scheme concocted. For simplicity, no compaction will be attempted so the states of 0 and 1 will be represented by a simple integer devoted to that bit.
The conversion scheme requires the values of the Fibonacci sequence, except not quite: the Fibonacci sequence starts F0 = 0, F1 = 1, F2 = 1, F3 = 2, F4 = 3, etc. but what is wanted is to start with the second 1, so F1 = 1, F2 = 2, F3 = 3, F4 = 5, etc. so this sequence has been named the Fib1nacci sequence to replace conceptual dissonance with lexical dissonance, and similarly with array F1B instead of FIB. Initial investigations show that F45 is the last before a thirty-two bit two's complement integer is overflowed, though systems offering INTEGER*8 could be pushed further.
Initialising this table in array F1B could be done via specifying the relevant values (computed separately, even by hand) or by some banal initialisation loop that would be executed on the first invocation of any of the routines requiring those values, a tedious and annoying rigmarole to organise. More interesting are the facilities offered by the PARAMETER statement (introduced with F77), with the further possibility that constants, so defined, would be held safe from accidental change, nor would there be initialising code to execute at run time. Alas, the obvious approach (commented out in the source) using array F1B is rejected by the F90 compiler even though it does allow a value to be defined in terms of other defined values, as is demonstrated by the horde of simple variables following. Despite being a multi-pass compiler, the dependencies will not be unravelled if the statements appear out of order. Fortran does not include a standard pre-processor stage, unlike say pl/i where it is built-in to the language and uses much the same syntax as normal pl/i statements, so loops, IF-tests and so forth are available. By such means, the upper limit of 45 could be determined, the initial values calculated, and the array be defined with initial values, all at compile time.
By declaring the horde of simple names to have the PRIVATE attribute, they will not litter the name space of routines invoking the module, but alas, they will still occupy their own storage space. Another possibility would be to use the EQUIVALENCE statement to have them placed within array F1B, but alas, as noted in 15_Puzzle_Game#Fortran, the compiler will not countenance PARAMETER statements for names engaged in such misbehviour. A pity.
Still another possibility would be to take advantage of the formula for calculating the values of the Fibonacci series directly (with careful attention to the offsets needed for the Fib1nacci sequence), but this formula is rather intimidating: <lang Fortran>F(N) = ((1 + SQRT(5))**N - (1 - SQRT(5))**N)/(SQRT(5)*2**N)</lang> It can easily be coded as a Fortran function (and would have to be double precision because 32-bit floating-point arithmetic is not accurate enough for integer constants approaching 32 bits), but alas, the compiler does not allow itself to take the risk of invoking a user-written function in a PARAMETER statement, even if the compiler had itself compiled it. For, who knows what it might do?
Given the array F1B the conversion from an integer to Zeckendorf digit sequence starts from the high-order end to find the highest F1B value not exceeding the number. There is a formula for this mentioned in the EchoLisp section, but it too is intimidating and the rounding of its result would also require checking. Rather than a linear search, a binary chop could be used, though at the cost of additional code. The location of the high-order digit is recorded on general principles, it being useful in formatting output for example. This requires a "first-time" test within the loop, that could be avoided if the conversion were to be done in two stages.
The special feature of the conversion lies in noting that F1B(n + 1) = F1B(n) + F1B(n - 1), the defining feature of the Fibonacci sequence. Thus, when a 1-bit is found (say it is bit n), the next bit down must be a 0 and so the test for it may be skipped, by incrementing L This is because if it were not 0, then the bit above (bit n + 1) would have been turned on in the previous stage instead. Because of this adjustment, the controlling loop cannot be DO L = ZLAST,1,-1 to step down the entries in array F1B because modifications to the index variable of a DO-loop, if not rejected out-of-hand by the compiler, may have no effect on the execution of the loop. This is because the execution of a DO-loop is often controlled by an "iteration count", calculated on entry to the DO-loop, which is thereby unaffected by changes to the index variable, or indeed to the bounds and step size of the loop. Other implementations of a DO-loop will offer other behaviour. There being no equivalent in Fortran of Repeat ... until ... ; (whereby the test is at the end, and there is no initial test), a GO TO appears...
The source uses F90 for its MODULE facility, in particular having array F1B available without having to mess about with additional parameters or COMMON statements. This also enables the specification of arrays with a lower bound other than one, which makes it easy to define the digit arrays to have a current length, stored in element zero. This sort of "string" facility is often restricted only to strings of characters, but the notion "string of <type>" is often useful. If in routines declared within a MODULE the size of an array parameter is declared via : there are secret additional parameters defining its size, accessible via special functions such as UBOUND so there is no need for an explicit parameter doing so as would be the case prior to F90. With F90 it is also possible to define a compound data type for the digit sequence, but a simple array seems more flexible.
The pleasing name, "MODULE ZECKENDORF ARITHMETIC" causes some odd behaviour, even though Fortran source normally involves spaces having no significance outside text literals.<lang Fortran> MODULE ZECKENDORF ARITHMETIC !Using the Fibonacci series, rather than powers of some base.
INTEGER ZLAST !The standard 32-bit two's complement integers
PARAMETER (ZLAST = 45) !only get so far, just as there's a limit to the highest power.
INTEGER F1B(ZLAST) !I want the Fibonacci series, and, starting with its second one.
c PARAMETER (F1B = (/1,2, !But alas, the compiler doesn't allow c 3 F1B(1) + F1B(2), !for this sort of carpet-unrolling c 4 F1B(2) + F1B(3), !initialisation sequence.
INTEGER,PRIVATE:: F01,F02,F03,F04,F05,F06,F07,F08,F09,F10, !So, not bothering with F00,
1 F11,F12,F13,F14,F15,F16,F17,F18,F19,F20, !Prepare a horde of simple names,
2 F21,F22,F23,F24,F25,F26,F27,F28,F29,F30, !which can be initialised
3 F31,F32,F33,F34,F35,F36,F37,F38,F39,F40, !in a certain way,
4 F41,F42,F43,F44,F45 !without scaring the compiler.
PARAMETER (F01 = 1, F02 = 2, F03 = F02 + F01, F04 = F03 + F02, !Thusly.
1 F05=F04+F03,F06=F05+F04,F07=F06+F05,F08=F07+F06,F09=F08+F07, !Typing all this
2 F10=F09+F08,F11=F10+F09,F12=F11+F10,F13=F12+F11,F14=F13+F12, !is an invitation
3 F15=F14+F13,F16=F15+F14,F17=F16+F15,F18=F17+F16,F19=F18+F17, !for mistypes.
4 F20=F19+F18,F21=F20+F19,F22=F21+F20,F23=F22+F21,F24=F23+F22, !So a regular layout
5 F25=F24+F23,F26=F25+F24,F27=F26+F25,F28=F27+F26,F29=F28+F27, !helps a little.
6 F30=F29+F28,F31=F30+F29,F32=F31+F30,F33=F32+F31,F34=F33+F32, !Otherwise,
7 F35=F34+F33,F36=F35+F34,F37=F36+F35,F38=F37+F36,F39=F38+F37, !devise a prog.
8 F40=F39+F38,F41=F40+F39,F42=F41+F40,F43=F42+F41,F44=F43+F42, !to generate these texts...
9 F45=F44+F43) !The next is 2971215073. Too big for 32-bit two's complement integers.
PARAMETER (F1B = (/F01,F02,F03,F04,F05,F06,F07,F08,F09,F10, !And now,
1 F11, F12, F13, F14, F15, F16, F17, F18, F19, F20, !Here is the desired
2 F21, F22, F23, F24, F25, F26, F27, F28, F29, F30, !array of constants.
3 F31, F32, F33, F34, F35, F36, F37, F38, F39, F40, !And as such, possibly
4 F41, F42, F43, F44, F45/)) !protected from alteration.
CONTAINS !After all that, here we go.
SUBROUTINE ZECK(N,D) !Convert N to a "Zeckendorf" digit sequence.
Counts upwards from digit one. D(i) ~ F1B(i). D(0) fingers the high-order digit.
INTEGER N !The normal number, in the computer's base.
INTEGER D(0:) !The digits, to be determined.
INTEGER R !The remnant.
INTEGER L !A finger, similar to the power of the base.
IF (N.LT.0) STOP "ZECK! No negative numbers!" !I'm not thinking about them.
R = N !Grab a copy that I can mess with.
D = 0 !Scrub the lot in one go.
L = ZLAST !As if starting with BASE**MAX, rather than BASE**0.
10 IF (R.GE.F1B(L)) THEN !Has the remnant sufficient for this digit?
R = R - F1B(L) !Yes! Remove that amount.
IF (D(0).EQ.0) THEN !Is this the first non-zero digit?
IF (L.GT.UBOUND(D,DIM=1)) STOP "ZECK! Not enough digits!" !Yes.
D(0) = L !Remember the location of the high-order digit.
END IF !Two loops instead, to avoid repeated testing?
D(L) = 1 !Place the digit, knowing a place awaits.
L = L - 1 !Never need a ...11... sequence because F1B(i) + F1B(i+1) = F1B(i+2).
END IF !So much for that digit "power".
L = L - 1 !Down a digit.
IF (L.GT.0 .AND. R.GT.0) GO TO 10 !Are we there yet?
IF (N.EQ.0) D(0) = 1 !Zero has one digit.
END SUBROUTINE ZECK !That was fun.
INTEGER FUNCTION ZECKN(D) !Converts a "Zeckendorf" digit sequence to a number.
INTEGER D(0:) !The digits. D(0) fingers the high-order digit.
IF (D(0).LE.0) STOP "ZECKN! Empty number!" !General paranoia.
IF (D(0).GT.ZLAST) STOP "ZECKN! Oversize number!" !I hate array bound hiccoughs.
ZECKN = SUM(D(1:D(0))*F1B(1:D(0))) !This is what positional notation means.
IF (ZECKN.LT.0) STOP "ZECKN! Integer overflow!" !Oh for IF OVERFLOW as in First Fortran.
END FUNCTION ZECKN !Overflows by a small amount will produce a negative number.
END MODULE ZECKENDORF ARITHMETIC !Odd stuff.
PROGRAM POKE
USE ZECKENDORF ARITHMETIC !Please.
INTEGER ZD(0:ZLAST) !A scratchpad.
INTEGER I,J,W
CHARACTER*1 DIGIT(0:1) !Assistance for the output.
PARAMETER (DIGIT = (/"0","1"/), W = 6) !This field width suffices.
c WRITE (6,*) F1B c WRITE (6,*) INT8(F1B(44)) + INT8(F1B(45))
WRITE (6,1) F1B(1:4),ZLAST,ZLAST,F1B(ZLAST),HUGE(I) !Show some provenance.
1 FORMAT ("Converts integers to their Zeckendorf digit string "
1 "using the Fib1nacci sequence (",4(I0,","),
2 " ...) as the equivalent of powers."/
3 "At most, ",I0," digits because Fib1nacci(",I0,") = ",I0,
4 " and the integer limit is ",I0,".",//," N ZN") !Ends with a heading.
DO I = 0,20 !Step through the specified range.
CALL ZECK(I,ZD) !Convert I to ZD.
c WRITE (6,2) I,ZD(ZD(0):1:-1) !Show digits from high-order to low. c 2 FORMAT (I3,1X,66I1) !Or, WRITE (6,2) I,(ZD(J), J = ZD(0),1,-1)
WRITE (6,3) I,(" ",J = ZD(0) + 1,W),DIGIT(ZD(ZD(0):1:-1)) !Right-aligned in field width W.
3 FORMAT (I3,1X,66A1) !The digits appear as characters.
IF (I.NE.ZECKN(ZD)) STOP "Huh?" !Should never happen...
END DO !On to the next.
END</lang>
Output: shown aligned right for a more regular table. Producing leading spaces or digits required a conversion from a numerical digit to a character digit, so that all the output could use the A format code.
Converts integers to their Zeckendorf digit string using the Fib1nacci sequence (1,2,3,5, ...) as the equivalent of powers. At most, 45 digits because Fib1nacci(45) = 1836311903 and the integer limit is 2147483647. N ZN 0 0 1 1 2 10 3 100 4 101 5 1000 6 1001 7 1010 8 10000 9 10001 10 10010 11 10100 12 10101 13 100000 14 100001 15 100010 16 100100 17 100101 18 101000 19 101001 20 101010
FreeBASIC
<lang freebasic>' version 17-10-2016 ' compile with: fbc -s console
- Define max 92 ' max for Fibonacci number
Dim Shared As ULongInt fib(max)
fib(0) = 1 fib(1) = 1
For x As Integer = 2 To max
fib(x) = fib(x-1) + fib(x-2)
Next
Function num2zeck(n As Integer) As String
If n < 0 Then
Print "Error: no negative numbers allowed" Beep : Sleep 5000,1 : End
End If
If n < 2 Then Return Str(n)
Dim As String zeckendorf
For x As Integer = max To 1 Step -1
If fib(x) <= n Then
zeckendorf = zeckendorf + "1"
n = n - fib(x)
Else
zeckendorf = zeckendorf + "0"
End If
Next
return LTrim(zeckendorf, "0") ' get rid of leading zeroes
End Function
' ------=< MAIN >=------
Dim As Integer x, e Dim As String zeckendorf Print "number zeckendorf"
For x = 0 To 200000
zeckendorf = num2zeck(x) If x <= 20 Then Print x, zeckendorf ' check for two consecutive Fibonacci numbers If InStr(zeckendorf, "11") <> 0 Then Print " Error: two consecutive Fibonacci numbers "; x, zeckendorf e = e +1 End If
Next
Print If e = 0 Then
Print " No Zeckendorf numbers with two consecutive Fibonacci numbers found"
Else
Print e; " error(s) found"
End If
' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End</lang>
- Output:
number zeckendorf 0 0 1 1 2 10 3 100 4 101 5 1000 6 1001 7 1010 8 10000 9 10001 10 10010 11 10100 12 10101 13 100000 14 100001 15 100010 16 100100 17 100101 18 101000 19 101001 20 101010 No Zeckendorf numbers with two consecutive Fibonacci numbers found
Go
<lang go>package main
import "fmt"
func main() {
for i := 0; i <= 20; i++ {
fmt.Printf("%2d %7b\n", i, zeckendorf(i))
}
}
func zeckendorf(n int) int {
// initial arguments of fib0 = 1 and fib1 = 1 will produce
// the Fibonacci sequence {1, 2, 3,..} on the stack as successive
// values of fib1.
_, set := zr(1, 1, n, 0)
return set
}
func zr(fib0, fib1, n int, bit uint) (remaining, set int) {
if fib1 > n {
return n, 0
}
// recurse.
// construct sequence on the way in, construct ZR on the way out.
remaining, set = zr(fib1, fib0+fib1, n, bit+1)
if fib1 <= remaining {
set |= 1 << bit
remaining -= fib1
}
return
}</lang>
- Output:
0 0 1 1 2 10 3 100 4 101 5 1000 6 1001 7 1010 8 10000 9 10001 10 10010 11 10100 12 10101 13 100000 14 100001 15 100010 16 100100 17 100101 18 101000 19 101001 20 101010
Haskell
Using "no consecutive 1s" rule: <lang haskell>import Data.Bits import Numeric
zeckendorf = map b $ filter ones [0..] where ones :: Int -> Bool ones x = 0 == x .&. (x `shiftR` 1) b x = showIntAtBase 2 ("01"!!) x ""
main = mapM_ putStrLn $ take 21 zeckendorf</lang> which is the same as <lang haskell>zeckendorf = "0":"1":[s++[d] | s <- tail zeckendorf, d <- "01", last s /= '1' || d /= '1']
main = mapM putStrLn $ take 21 zeckendorf</lang> or a different way to generate the sequence: <lang haskell>import Numeric
fib = 1 : 1 : zipWith (+) fib (tail fib) pow2 = iterate (2*) 1
zeckendorf = map b z where z = 0:concat (zipWith f fib pow2) f x y = map (y+) (take x z) b x = showIntAtBase 2 ("01"!!) x ""
main = mapM_ putStrLn $ take 21 zeckendorf</lang>
Creating a string for an individual number: <lang haskell>import Data.List (mapAccumL)
fib :: [Int] fib = 1 : 2 : zipWith (+) fib (tail fib)
zeckendorf :: Int -> String zeckendorf 0 = "0" zeckendorf n = snd $ mapAccumL f n $ reverse $ takeWhile (<= n) fib
where
f n x
| n < x = (n, '0')
| otherwise = (n - x, '1')
main :: IO () main = (putStrLn . unlines) $ zeckendorf <$> [0 .. 20]</lang>
- Output:
0 1 10 100 101 1000 1001 1010 10000 10001 10010 10100 10101 100000 100001 100010 100100 100101 101000 101001 101010
J
Please enjoy our Zeckendorf essay. <lang J> fib=: 3 : 0 " 0
mp=. +/ .*
{.{: mp/ mp~^:(I.|.#:y) 2 2$0 1 1 1x
)
phi=: -:1+%:5
fi =: 3 : 'n - y<fib n=. 0>.(1=y)-~>.(phi^.%:5)+phi^.y'
fsum=: 3 : 0
z=. 0$r=. y while. 3<r do. m=. fib fi r z=. z,m r=. r-m end. z,r$~(*r)+.0=y
)
Filter=: (#~`)(`:6)
' '&~:Filter@:":@:#:@:#.@:((|. fib 2+i.8) e. fsum)&.>i.3 7 ┌──────┬──────┬──────┬──────┬──────┬──────┬──────┐ │0 │1 │10 │100 │101 │1000 │1001 │ ├──────┼──────┼──────┼──────┼──────┼──────┼──────┤ │1010 │10000 │10001 │10010 │10100 │10101 │100000│ ├──────┼──────┼──────┼──────┼──────┼──────┼──────┤ │100001│100010│100100│100101│101000│101001│101010│ └──────┴──────┴──────┴──────┴──────┴──────┴──────┘ </lang>
Explanation:
fsum finds the canonical list of fibonacci terms which sum to its argument.
fib finds the nth fibonacci term of the fibonacci sequence. This would be 0 1 1 2 3 5 8 13 21 34 55 89 ... but we ignore the first two values of that sequence for the purpose of this exercise.
(|. fib 2+i.8) is 34 21 13 8 5 3 2 1. You can think of an eight bit Zeckendorf number such as 101010 as representing the inner product of its digits with (|. fib 2+i.8). Thus, we can find the relevant Zeckendorf bits by finding which which members of that sequence are in the result of fsum
The rest is just formatting. (We convert from binary list to integer and then back to binary list, to eliminate leading zeros from the list. Then we convert to text and remove all the spaces. Since we arranged for each result to be in a box, the boxes will align giving us a somewhat readable presentation.
Java
Code:
<lang java>import java.util.*;
class Zeckendorf {
public static String getZeckendorf(int n)
{
if (n == 0)
return "0";
List<Integer> fibNumbers = new ArrayList<Integer>();
fibNumbers.add(1);
int nextFib = 2;
while (nextFib <= n)
{
fibNumbers.add(nextFib);
nextFib += fibNumbers.get(fibNumbers.size() - 2);
}
StringBuilder sb = new StringBuilder();
for (int i = fibNumbers.size() - 1; i >= 0; i--)
{
int fibNumber = fibNumbers.get(i);
sb.append((fibNumber <= n) ? "1" : "0");
if (fibNumber <= n)
n -= fibNumber;
}
return sb.toString();
}
public static void main(String[] args)
{
for (int i = 0; i <= 20; i++)
System.out.println("Z(" + i + ")=" + getZeckendorf(i));
}
}</lang>
Output:
Z(0)=0 Z(1)=1 Z(2)=10 Z(3)=100 Z(4)=101 Z(5)=1000 Z(6)=1001 Z(7)=1010 Z(8)=10000 Z(9)=10001 Z(10)=10010 Z(11)=10100 Z(12)=10101 Z(13)=100000 Z(14)=100001 Z(15)=100010 Z(16)=100100 Z(17)=100101 Z(18)=101000 Z(19)=101001 Z(20)=101010
Recursive Implementation
Code: <lang java>import java.util.ArrayList; import java.util.List;
public class Zeckendorf {
private List<Integer> getFibList(final int maxNum, final int n1, final int n2, final List<Integer> fibs){
if(n2 > maxNum) return fibs;
fibs.add(n2);
return getFibList(maxNum, n2, n1 + n2, fibs); }
public String getZeckendorf(final int num) {
if (num <= 0) return "0";
final List<Integer> fibs = getFibList(num, 1, 2, new ArrayList<Integer>()Template:Add(1););
return getZeckString("", num, fibs.size() - 1, fibs);
}
private String getZeckString(final String zeck, final int num, final int index, final List<Integer> fibs){
final int curFib = fibs.get(index);
final boolean placeZeck = num >= curFib;
final String outString = placeZeck ? zeck + "1" : zeck + "0";
final int outNum = placeZeck ? num - curFib : num;
if(index == 0) return outString;
return getZeckString(outString, outNum, index - 1, fibs);
}
public static void main(final String[] args) {
final Zeckendorf zeckendorf = new Zeckendorf();
for(int i =0; i <= 20; i++){
System.out.println("Z("+ i +"):\t" + zeckendorf.getZeckendorf(i));
}
}
}</lang>
Output:
Z(0): 0 Z(1): 1 Z(2): 10 Z(3): 100 Z(4): 101 Z(5): 1000 Z(6): 1001 Z(7): 1010 Z(8): 10000 Z(9): 10001 Z(10): 10010 Z(11): 10100 Z(12): 10101 Z(13): 100000 Z(14): 100001 Z(15): 100010 Z(16): 100100 Z(17): 100101 Z(18): 101000 Z(19): 101001 Z(20): 101010
JavaScript
ES6
(mapAccumuL example)
<lang JavaScript>(() => {
'use strict';
const main = () =>
unlines(
map(n => concat(zeckendorf(n)),
enumFromTo(0, 20)
)
);
// zeckendorf :: Int -> String
const zeckendorf = n => {
const go = (n, x) =>
n < x ? (
Tuple(n, '0')
) : Tuple(n - x, '1')
return 0 < n ? (
snd(mapAccumL(
go, n,
reverse(fibUntil(n))
))
) : ['0'];
};
// fibUntil :: Int -> [Int]
const fibUntil = n =>
cons(1, takeWhile(x => n >= x,
map(snd, iterateUntil(
tpl => n <= fst(tpl),
tpl => {
const x = snd(tpl);
return Tuple(x, x + fst(tpl));
},
Tuple(1, 2)
))));
// GENERIC FUNCTIONS ----------------------------
// Tuple (,) :: a -> b -> (a, b)
const Tuple = (a, b) => ({
type: 'Tuple',
'0': a,
'1': b,
length: 2
});
// concat :: a -> [a] // concat :: [String] -> String const concat = xs => 0 < xs.length ? (() => { const unit = 'string' !== typeof xs[0] ? ( [] ) : ; return unit.concat.apply(unit, xs); })() : [];
// cons :: a -> [a] -> [a]
const cons = (x, xs) =>
Array.isArray(xs) ? (
[x].concat(xs)
) : (x + xs);
// enumFromTo :: Int -> Int -> [Int]
const enumFromTo = (m, n) =>
m <= n ? iterateUntil(
x => n <= x,
x => 1 + x,
m
) : [];
// fst :: (a, b) -> a const fst = tpl => tpl[0];
// iterateUntil :: (a -> Bool) -> (a -> a) -> a -> [a]
const iterateUntil = (p, f, x) => {
const vs = [x];
let h = x;
while (!p(h))(h = f(h), vs.push(h));
return vs;
};
// map :: (a -> b) -> [a] -> [b] const map = (f, xs) => xs.map(f);
// 'The mapAccumL function behaves like a combination of map and foldl; // it applies a function to each element of a list, passing an accumulating // parameter from left to right, and returning a final value of this // accumulator together with the new list.' (See Hoogle)
// mapAccumL :: (acc -> x -> (acc, y)) -> acc -> [x] -> (acc, [y])
const mapAccumL = (f, acc, xs) =>
xs.reduce((a, x, i) => {
const pair = f(a[0], x, i);
return Tuple(pair[0], a[1].concat(pair[1]));
}, Tuple(acc, []));
// reverse :: [a] -> [a]
const reverse = xs =>
'string' !== typeof xs ? (
xs.slice(0).reverse()
) : xs.split().reverse().join();
// snd :: (a, b) -> b const snd = tpl => tpl[1];
// tail :: [a] -> [a] const tail = xs => 0 < xs.length ? xs.slice(1) : [];
// takeWhile :: (a -> Bool) -> [a] -> [a]
// takeWhile :: (Char -> Bool) -> String -> String
const takeWhile = (p, xs) => {
const lng = xs.length;
return 0 < lng ? xs.slice(
0,
until(
i => i === lng || !p(xs[i]),
i => 1 + i,
0
)
) : [];
};
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// until :: (a -> Bool) -> (a -> a) -> a -> a
const until = (p, f, x) => {
let v = x;
while (!p(v)) v = f(v);
return v;
};
// MAIN --- return main();
})();</lang>
- Output:
0 1 10 100 101 1000 1001 1010 10000 10001 10010 10100 10101 100000 100001 100010 100100 100101 101000 101001 101010
jq
<lang jq>def zeckendorf:
def fibs($n): # input: [f(i-2), f(i-1)] [1,1] | [recurse(select(.[1] < $n) | [.[1], add]) | .[1]] ;
# Emit an array of 0s and 1s corresponding to the Zeckendorf encoding
# $f should be the relevant Fibonacci numbers in increasing order.
def loop($f):
[ recurse(. as [$n, $ix]
| select( $ix > -1 )
| $f[$ix] as $next
| if $n >= $next
then [$n - $next, $ix-1, 1]
else [$n, $ix-1, 0]
end )
| .[2] // empty ]
# remove any superfluous leading 0:
# remove leading 0 if any unless length==1
| if length>1 and .[0] == 0 then .[1:] else . end ;
# state: [$n, index_in_fibs, digit ]
fibs(.) as $f
| [., ($f|length)-1]
| loop($f)
| join("") ;</lang>
Example: <lang jq>range(0;21) | "\(.): \(zeckendorf)"</lang>
- Output:
<lang sh>$ jq -n -r -f zeckendorf.jq 0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010</lang>
Julia
<lang julia>function zeck(n)
n <= 0 && return 0 fib = [2,1]; while fib[1] < n unshift!(fib,sum(fib[1:2])) end dig = Int[]; for f in fib f <= n ? (push!(dig,1); n = n-f;) : push!(dig,0) end return dig[1] == 0 ? dig[2:end] : dig
end</lang>
- Output:
julia> for x = 0:20
println(join(zeck(x)))
end
0
1
10
100
101
1000
1001
1010
10000
10001
10010
10100
10101
100000
100001
100010
100100
100101
101000
101001
101010
Klingphix
<lang Klingphix>include ..\Utilitys.tlhy
- listos
%i$ "" !i$ len [ get tostr $i$ chain !i$ ] for drop $i$
- Zeckendorf %n !n
%i 0 !i %c 0 !c
[
$i 8 itob listos
"11" find not (
[ ( $c ":" 9 tochar ) lprint tonum ? $c 1 + !c ]
[drop]
) if
$i 1 + !i
]
[$c $n >] until
20 Zeckendorf
nl "End " input</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010 End
Kotlin
<lang scala>// version 1.0.6
const val LIMIT = 46 // to stay within range of signed 32 bit integer val fibs = fibonacci(LIMIT)
fun fibonacci(n: Int): IntArray {
if (n !in 2..LIMIT) throw IllegalArgumentException("n must be between 2 and $LIMIT")
val fibs = IntArray(n)
fibs[0] = 1
fibs[1] = 1
for (i in 2 until n) fibs[i] = fibs[i - 1] + fibs[i - 2]
return fibs
}
fun zeckendorf(n: Int): String {
if (n < 0) throw IllegalArgumentException("n must be non-negative")
if (n < 2) return n.toString()
var lastFibIndex = 1
for (i in 2..LIMIT)
if (fibs[i] > n) {
lastFibIndex = i - 1
break
}
var nn = n - fibs[lastFibIndex--]
val zr = StringBuilder("1")
for (i in lastFibIndex downTo 1)
if (fibs[i] <= nn) {
zr.append('1')
nn -= fibs[i]
} else {
zr.append('0')
}
return zr.toString()
}
fun main(args: Array<String>) {
println(" n z")
for (i in 0..20) println("${"%2d".format(i)} : ${zeckendorf(i)}")
}</lang>
- Output:
n z 0 : 0 1 : 1 2 : 10 3 : 100 4 : 101 5 : 1000 6 : 1001 7 : 1010 8 : 10000 9 : 10001 10 : 10010 11 : 10100 12 : 10101 13 : 100000 14 : 100001 15 : 100010 16 : 100100 17 : 100101 18 : 101000 19 : 101001 20 : 101010
Liberty BASIC
CBTJD: 2020/03/09 <lang vb>samples = 20 call zecklist samples
print "Decimal","Zeckendorf" for n = 0 to samples
print n, zecklist$(n)
next n
Sub zecklist inDEC
dim zecklist$(inDEC)
do
bin$ = dec2bin$(count)
if instr(bin$,"11") = 0 then
zecklist$(found) = bin$
found = found + 1
end if
count = count+1
loop until found = inDEC + 1
End sub
function dec2bin$(inDEC)
do bin$ = str$(inDEC mod 2) + bin$ inDEC = int(inDEC/2) loop until inDEC = 0 dec2bin$ = bin$
end function</lang>
Lingo
<lang Lingo>-- Return the distinct Fibonacci numbers not greater than 'n' on fibsUpTo (n)
fibList = []
last = 1
current = 1
repeat while current <= n
fibList.add(current)
nxt = last + current
last = current
current = nxt
end repeat
return fibList
end
-- Return the Zeckendorf representation of 'n' on zeckendorf (n)
fib = fibsUpTo(n)
zeck = ""
repeat with pos = fib.count down to 1
if n >= fib[pos] then
zeck = zeck & "1"
n = n - fib[pos]
else
zeck = zeck & "0"
end if
end repeat
if zeck = "" then return "0"
return zeck
end</lang>
<lang Lingo>repeat with n = 0 to 20
put n & ": " & zeckendorf(n)
end repeat</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Little Man Computer
Runs in Peter Higginson's LMC simulator. Uses his non-standard OTC instruction to output the character whose ASCII code is in the accumulator.
Most online LMC simulators seem to have very limited space for output. Peter Higginson's allows only sixteen lines of four characters each. Previous output scrolls off and is lost. The output from this program had to be captured by repeatedly pausing the program and using copy-and-paste. <lang Little Man Computer> // Little Man Computer, for Rosetta Code. // Writes Zeckendorf representations of numbers 0..20. // Works with Peter Higginson's LMC simulator, except that // user must intervene manually to capture all the output.
LDA c0 // initialize to N = 0
loop STA N
OUT // write N
LDA equals // then equals sign
OTC
BRA wr_zeck // then Zeckendorf rep
return LDA space // then space
OTC
LDA N // done maximum N?
SUB N_max
BRZ halt // yes, halt
LDA N // no, inc N and loop back
ADD c1
BRA loop
halt HLT c0 DAT 0 N_max DAT 20 equals DAT 61 space DAT 32
// Routine to write Zeckendorf representation of number stored in N. // Since LMC doesn't support subroutines, returns with "BRA return". wr_zeck LDA N
SUB c1
BRP phase_1
// N = 0, special case
LDA ascii_0
OTC
BRA done
// N > 0. Phase 1: find largest Fibonacci number <= N phase_1 STA res // res := N - 1
LDA c1 // initialize Fibonacci terms
STA a
STA b
loop_1 LDA res // here res = N - a (easy proof)
SUB b // is next Fibonacci a + b > N?
BRP next_fib // no, continue Fibonacci
BRA phase_2 // yes, on to phase 2
next_fib STA res // res := res - b
LDA a // (a, b) := (a + b, a)
ADD b
STA a
SUB b
STA b
BRA loop_1 // loop to test new (a, b)
// Phase 2: get Zeckendorf digits by winding Fibonacci back phase_2 LDA ascii_1 // first digit must be 1
OTC
loop_2 LDA a // done when wound back to a = 1
SUB c1
BRZ done
LDA res // decide next Zeckendorf digit
SUB b // 0 if res < b, 1 if res >= b
BRP dig_is_1
LDA ascii_0
BRA wr_dig
dig_is_1 STA res // res := res - b
LDA ascii_1
wr_dig OTC // write Zeckendorf digit 0 or 1
LDA a // (a, b) := (b, a - b)
SUB b
STA b
LDA a
SUB b
STA a
BRA loop_2 // loop to test new (a, b)
done BRA return N DAT res DAT a DAT b DAT c1 DAT 1 ascii_0 DAT 48 ascii_1 DAT 49 // end </lang>
- Output:
[formatted manually] 0=0 1=1 2=10 3=100 4=101 5=1000 6=1001 7=1010 8=10000 9=10001 10=10010 11=10100 12=10101 13=100000 14=100001 15=100010 16=100100 17=100101 18=101000 19=101001 20=101010
Logo
<lang logo>; return the (N+1)th Fibonacci number (1,2,3,5,8,13,...) to fib m
local "n make "n sum :m 1 if [lessequal? :n 0] [output difference fib sum :n 2 fib sum :n 1] global "_fib if [not name? "_fib] [ make "_fib [1 1] ] local "length make "length count :_fib while [greater? :n :length] [ make "_fib (lput (sum (last :_fib) (last (butlast :_fib))) :_fib) make "length sum :length 1 ] output item :n :_fib
end
- return the binary Zeckendorf representation of a nonnegative number
to zeckendorf n
if [less? :n 0] [(throw "error [Number must be nonnegative.])] (local "i "f "result) make "i :n make "f fib :i while [less? :f :n] [make "i sum :i 1 make "f fib :i]
make "result "||
while [greater? :i 0] [
ifelse [greaterequal? :n :f] [
make "result lput 1 :result
make "n difference :n :f
] [
if [not empty? :result] [
make "result lput 0 :result
]
]
make "i difference :i 1
make "f fib :i
]
if [equal? :result "||] [
make "result 0
]
output :result
end
type zeckendorf 0 repeat 20 [
type word "| | zeckendorf repcount
] print [] bye</lang>
- Output:
0 1 10 100 101 1000 1001 1010 10000 10001 10010 10100 10101 100000 100001 100010 100100 100101 101000 101001 101010
Lua
<lang Lua>-- Return the distinct Fibonacci numbers not greater than 'n' function fibsUpTo (n)
local fibList, last, current, nxt = {}, 1, 1
while current <= n do
table.insert(fibList, current)
nxt = last + current
last = current
current = nxt
end
return fibList
end
-- Return the Zeckendorf representation of 'n' function zeckendorf (n)
local fib, zeck = fibsUpTo(n), ""
for pos = #fib, 1, -1 do
if n >= fib[pos] then
zeck = zeck .. "1"
n = n - fib[pos]
else
zeck = zeck .. "0"
end
end
if zeck == "" then return "0" end
return zeck
end
-- Main procedure print(" n\t| Zeckendorf(n)") print(string.rep("-", 23)) for n = 0, 20 do
print(" " .. n, "| " .. zeckendorf(n))
end</lang>
- Output:
n | Zeckendorf(n) ----------------------- 0 | 0 1 | 1 2 | 10 3 | 100 4 | 101 5 | 1000 6 | 1001 7 | 1010 8 | 10000 9 | 10001 10 | 10010 11 | 10100 12 | 10101 13 | 100000 14 | 100001 15 | 100010 16 | 100100 17 | 100101 18 | 101000 19 | 101001 20 | 101010
Mathematica/Wolfram Language
<lang Mathematica>zeckendorf[0] = 0; zeckendorf[n_Integer] :=
10^(# - 1) + zeckendorf[n - Fibonacci[# + 1]] &@
LengthWhile[
Fibonacci /@
Range[2, Ceiling@Log[GoldenRatio, n Sqrt@5]], # <= n &];
zeckendorf /@ Range[0, 20]</lang>
- Output:
{0, 1, 10, 100, 101, 1000, 1001, 1010, 10000, 10001, 10010, 10100,
10101, 100000, 100001, 100010, 100100, 100101, 101000, 101001, 101010}
Nim
<lang nim>import strformat, strutils
proc z(n: Natural): string =
if n == 0: return "0"
var fib = @[2,1]
var n = n
while fib[0] < n: fib.insert(fib[0] + fib[1])
for f in fib:
if f <= n:
result.add '1'
dec n, f
else:
result.add '0'
if result[0] == '0':
result = result[1..result.high]
for i in 0 .. 20:
echo &"{i:>3} {z(i):>8}"</lang>
- Output:
0 0 1 1 2 10 3 100 4 101 5 1000 6 1001 7 1010 8 10000 9 10001 10 10010 11 10100 12 10101 13 100000 14 100001 15 100010 16 100100 17 100101 18 101000 19 101001 20 101010
PARI/GP
<lang parigp>Z(n)=if(!n,print1(0));my(k=2);while(fibonacci(k)<=n,k++); forstep(i=k-1,2,-1,print1(if(fibonacci(i)<=n,n-=fibonacci(i);1,0)));print for(n=0,20,Z(n))</lang>
0 1 10 100 101 1000 1001 1010 10000 10001 10010 10100 10101 100000 100001 100010 100100 100101 101000 101001 101010
Pascal
A console application in Free Pascal, created with the Lazarus IDE. Though written independently of the Tcl solution, it uses essentially the same algorithm. <lang pascal> program ZeckendorfRep_RC;
{$mode objfpc}{$H+}
uses SysUtils;
// Return Zeckendorf representation of the passed-in cardinal. function ZeckRep( C : cardinal) : string; var
a, b, rem : cardinal; j, nrDigits: integer;
begin
// Case C = 0 has to be treated specially
if (C = 0) then begin
result := '0';
exit;
end;
// Find largest Fibonacci number not exceeding C
a := 1;
b := 1;
nrDigits := 1;
rem := C - 1;
while (rem >= b) do begin
dec( rem, b);
inc( a, b);
b := a - b;
inc( nrDigits);
end;
// Fill in digits by reversing Fibonacci back to start
SetLength( result, nrDigits);
j := 1;
result[j] := '1';
for j := 2 to nrDigits do begin
if (rem >= b) then begin
dec( rem, b);
result[j] := '1';
end
else result[j] := '0';
b := a - b;
dec( a, b);
end;
// Assert((a = 1) and (b = 1)); // optional check end;
// Main routine var
C : cardinal;
begin
for C := 1 to 20 do WriteLn( SysUtils.Format( '%2d: %s', [C, ZeckRep(C)]));
end. </lang>
- Output:
1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Perl
<lang perl>my @fib;
sub fib {
my $n = shift; return 1 if $n < 2; return $fib[$n] //= fib($n-1)+fib($n-2);
}
sub zeckendorf {
my $n = shift;
return "0" unless $n;
my $i = 1;
$i++ while fib($i) <= $n;
my $z = ;
while( --$i ) {
$z .= "0", next if fib( $i ) > $n;
$z .= "1";
$n -= fib( $i );
}
return $z;
}
printf "%4d: %8s\n", $_, zeckendorf($_) for 0..20; </lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Phix
function zeckendorf(integer n) integer r = 0, c sequence fib = {1,1} while fib[$]<n do fib &= fib[$] + fib[$-1] end while for i=length(fib) to 2 by -1 do c = n>=fib[i] r += r+c n -= c*fib[i] end for return r end function for i=0 to 20 do printf(1,"%2d: %7b\n",{i,zeckendorf(i)}) end for
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Phixmonti
<lang Phixmonti>def Zeckendorf /# n -- #/
0 var i 0 var c 1 1 2 tolist var pattern
true
while
i 8 int>bit reverse
pattern find
not if
c print ":\t" print print nl
dup c == if
false
else
c 1 + var c
true
endif
endif
i 1 + var i
endwhile
drop
enddef
20 Zeckendorf</lang>
PHP
<lang PHP> <?php $m = 20;
$F = array(1,1); while ($F[count($F)-1] <= $m)
$F[] = $F[count($F)-1] + $F[count($F)-2];
while ($n = $m--) {
while ($F[count($F)-1] > $n) array_pop($F);
$l = count($F)-1;
print "$n: ";
while ($n) {
if ($n >= $F[$l]) {
$n = $n - $F[$l];
print '1';
} else print '0';
--$l;
}
print str_repeat('0',$l);
print "\n";
} ?> </lang>
- Output:
20: 101010 19: 101001 18: 101000 17: 100101 16: 100100 15: 100010 14: 100001 13: 100000 12: 10101 11: 10100 10: 10010 9: 10001 8: 10000 7: 1010 6: 1001 5: 1000 4: 101 3: 100 2: 10 1: 1
PicoLisp
<lang PicoLisp>(de fib (N)
(let Fibs (1 1)
(while (>= N (+ (car Fibs) (cadr Fibs)))
(push 'Fibs (+ (car Fibs) (cadr Fibs))) )
(uniq Fibs) ) )
(de zecken1 (N)
(make
(for I (fib N)
(if (> I N)
(link 0)
(link 1)
(dec 'N I) ) ) ) )
(de zecken2 (N)
(make
(when (=0 N) (link 0))
(for I (fib N)
(when (<= I N)
(link I)
(dec 'N I) ) ) ) )
(for (N 0 (> 21 N) (inc N))
(tab (2 4 6 2 -10)
N
" -> "
(zecken1 N)
" "
(glue " + " (zecken2 N)) ) )
(bye)</lang>
- Output:
0 -> 0 0 1 -> 1 1 2 -> 10 2 3 -> 100 3 4 -> 101 3 + 1 5 -> 1000 5 6 -> 1001 5 + 1 7 -> 1010 5 + 2 8 -> 10000 8 9 -> 10001 8 + 1 10 -> 10010 8 + 2 11 -> 10100 8 + 3 12 -> 10101 8 + 3 + 1 13 -> 100000 13 14 -> 100001 13 + 1 15 -> 100010 13 + 2 16 -> 100100 13 + 3 17 -> 100101 13 + 3 + 1 18 -> 101000 13 + 5 19 -> 101001 13 + 5 + 1 20 -> 101010 13 + 5 + 2
Plain TeX
This code needs an etex engine.
<lang tex>\def\genfibolist#1{% #creates the fibo list which sum>=#1 \let\fibolist\empty\def\targetsum{#1}\def\fibosum{0}% \genfibolistaux1,1\relax } \def\genfibolistaux#1,#2\relax{% \ifnum\fibosum<\targetsum\relax \edef\fibosum{\number\numexpr\fibosum+#2}% \edef\fibolist{#2,\fibolist}% \edef\tempfibo{\noexpand\genfibolistaux#2,\number\numexpr#1+#2\relax\relax}% \expandafter\tempfibo \fi } \def\zeckendorf#1{\expandafter\zeckendorfaux\fibolist,\relax#1\relax\relax0} \def\zeckendorfaux#1,#2\relax#3\relax#4\relax#5{% \ifx\relax#2\relax #4% \else \ifnum#3<#1 \edef\temp{#2\relax#3\relax#4\ifnum#5=1 0\fi\relax#5}% \else \edef\temp{#2\relax\number\numexpr#3-#1\relax\relax#41\relax1}% \fi \expandafter\expandafter\expandafter\zeckendorfaux\expandafter\temp \fi } \newcount\ii \def\listzeckendorf#1{% \genfibolist{#1}% \ii=0 \loop \ifnum\ii<#1 \advance\ii1 \number\ii: \zeckendorf\ii\endgraf \repeat } \listzeckendorf{20}% any integer accepted \bye</lang>
pdf output looks like:
1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
PowerShell
<lang PowerShell> function Get-ZeckendorfNumber ( $N )
{
# Calculate relevant portation of Fibonacci series
$Fib = @( 1, 1 )
While ( $Fib[-1] -lt $N ) { $Fib += $Fib[-1] + $Fib[-2] }
# Start with 0
$ZeckendorfNumber = 0
# For each number in the relevant portion of Fibonacci series
For ( $i = $Fib.Count - 1; $i -gt 0; $i-- )
{
# If Fibonacci number is less than or equal to remainder of N
If ( $Fib[$i] -le $N )
{
# Double Z number and add 1 (equivalent to adding a '1' to the end of a binary number)
$ZeckendorfNumber = $ZeckendorfNumber * 2 + 1
# Reduce N by Fibonacci number, skip next Fibonacci number
$N -= $Fib[$i--]
}
# If were aren't finished yet, double Z number
# (equivalent to adding a '0' to the end of a binary number)
If ( $i ) { $ZeckendorfNumber *= 2 }
}
return $ZeckendorfNumber
}
</lang> <lang PowerShell>
- Get Zeckendorf numbers through 20, convert to binary for display
0..20 | ForEach { [convert]::ToString( ( Get-ZeckendorfNumber $_ ), 2 ) } </lang>
- Output:
0 1 10 100 101 1000 1001 1010 10000 10001 10010 10100 10101 100000 100001 100010 100100 100101 101000 101001 101010
PureBasic
<lang PureBasic>Procedure.s zeck(n.i)
Dim f.i(1) : Define i.i=1, o$ f(0)=1 : f(1)=1 While f(i)<n i+1 : ReDim f(ArraySize(f())+1) : f(i)=f(i-1)+f(i-2) Wend For i=i To 1 Step -1 If n>=f(i) : o$+"1" : n-f(i) : Else : o$+"0" : EndIf Next If Len(o$)>1 : o$=LTrim(o$,"0") : EndIf ProcedureReturn o$
EndProcedure
Define n.i, t$ OpenConsole("Zeckendorf number representation") PrintN(~"\tNr.\tZeckendorf") For n=0 To 20
t$=zeck(n)
If FindString(t$,"11")
PrintN("Error: n= "+Str(n)+~"\tZeckendorf= "+t$)
Break
Else
PrintN(~"\t"+RSet(Str(n),3," ")+~"\t"+RSet(t$,7," "))
EndIf
Next Input()</lang>
- Output:
Nr. Zeckendorf
0 0
1 1
2 10
3 100
4 101
5 1000
6 1001
7 1010
8 10000
9 10001
10 10010
11 10100
12 10101
13 100000
14 100001
15 100010
16 100100
17 100101
18 101000
19 101001
20 101010
Python
<lang python>def fib():
memo = [1, 2]
while True:
memo.append(sum(memo))
yield memo.pop(0)
def sequence_down_from_n(n, seq_generator):
seq = []
for s in seq_generator():
seq.append(s)
if s >= n: break
return seq[::-1]
def zeckendorf(n):
if n == 0: return [0]
seq = sequence_down_from_n(n, fib)
digits, nleft = [], n
for s in seq:
if s <= nleft:
digits.append(1)
nleft -= s
else:
digits.append(0)
assert nleft == 0, 'Check all of n is accounted for'
assert sum(x*y for x,y in zip(digits, seq)) == n, 'Assert digits are correct'
while digits[0] == 0:
# Remove any zeroes padding L.H.S.
digits.pop(0)
return digits
n = 20 print('Fibonacci digit multipliers: %r' % sequence_down_from_n(n, fib)) for i in range(n + 1):
print('%3i: %8s' % (i, .join(str(d) for d in zeckendorf(i))))</lang>
- Output:
Fibonacci digit multipliers: [21, 13, 8, 5, 3, 2, 1] 0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Shorter version
<lang python>n = 20 def z(n):
if n == 0 : return [0]
fib = [2,1]
while fib[0] < n: fib[0:0] = [sum(fib[:2])]
dig = []
for f in fib:
if f <= n:
dig, n = dig + [1], n - f
else:
dig += [0]
return dig if dig[0] else dig[1:]
for i in range(n + 1):
print('%3i: %8s' % (i, .join(str(d) for d in z(i))))</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Quackery
<lang Quackery> [ 2 base put
echo base release ] is binecho ( n --> )
[ 0 swap ' [ 2 1 ]
[ 2dup 0 peek < iff
[ behead drop ]
done
dup 0 peek
over 1 peek
+ swap join again ]
witheach
[ rot 1 << unrot
2dup < iff drop
else
[ -
dip
[ 1 | ] ] ]
drop ] is n->z ( n --> z )
21 times
[ i^ dup 10 < if sp
dup echo sp
n->z binecho cr ]</lang>
- Output:
0 0 1 1 2 10 3 100 4 101 5 1000 6 1001 7 1010 8 10000 9 10001 10 10010 11 10100 12 10101 13 100000 14 100001 15 100010 16 100100 17 100101 18 101000 19 101001 20 101010
R
<lang R>zeckendorf <- function(number) {
# Get an upper limit on Fibonacci numbers needed to cover number
indexOfFibonacciNumber <- function(n) {
if (n < 1) {
2
} else {
Phi <- (1 + sqrt(5)) / 2
invertClosedFormula <- log(n * sqrt(5)) / log(Phi)
ceiling(invertClosedFormula)
}
}
upperLimit <- indexOfFibonacciNumber(number)
# Return the sequence as digits, sorted descending
fibonacciSequenceDigits <- function(n) {
fibGenerator <- function(f, ...) { c(f[2], sum(f)) }
fibSeq <- Reduce(fibGenerator, 1:n, c(0,1), accumulate=TRUE)
fibNums <- unlist(lapply(fibSeq, head, n=1))
# drop last F0 and F1 and reverse sequence rev(fibNums[-2:-1]) }
digits <- fibonacciSequenceDigits(upperLimit)
isInNumber <- function(digit) {
if (number >= digit) {
number <<- number - digit
1
} else {
0
}
}
zeckSeq <- Map(isInNumber, digits)
# drop leading 0 and convert to String
gsub("^0+1", "1", paste(zeckSeq, collapse=""))
}
print(unlist(lapply(0:20, zeckendorf)))</lang>
This is definitely not the shortest way to implement the Zeckendorf numbers but focus was on the functional aspect of R, so no loops and (almost) no assignments.
- Output:
[1] "0" "1" "10" "100" "101" "1000" "1001" "1010" [9] "10000" "10001" "10010" "10100" "10101" "100000" "100001" "100010" [17] "100100" "100101" "101000" "101001" "101010"
Racket
<lang racket>
- lang racket (require math)
(define (fibs n)
(reverse
(for/list ([i (in-naturals 2)] #:break (> (fibonacci i) n))
(fibonacci i))))
(define (zechendorf n)
(match/values
(for/fold ([n n] [xs '()]) ([f (fibs n)])
(if (> f n)
(values n (cons 0 xs))
(values (- n f) (cons 1 xs))))
[(_ xs) (reverse xs)]))
(for/list ([n 21])
(list n (zechendorf n)))
</lang> Output: <lang racket> '((0 ())
(1 (1)) (2 (1 0)) (3 (1 0 0)) (4 (1 0 1)) (5 (1 0 0 0)) (6 (1 0 0 1)) (7 (1 0 1 0)) (8 (1 0 0 0 0)) (9 (1 0 0 0 1)) (10 (1 0 0 1 0)) (11 (1 0 1 0 0)) (12 (1 0 1 0 1)) (13 (1 0 0 0 0 0)) (14 (1 0 0 0 0 1)) (15 (1 0 0 0 1 0)) (16 (1 0 0 1 0 0)) (17 (1 0 0 1 0 1)) (18 (1 0 1 0 0 0)) (19 (1 0 1 0 0 1)) (20 (1 0 1 0 1 0)))
</lang>
Raku
(formerly Perl 6)
<lang perl6>printf "%2d: %8s\n", $_, zeckendorf($_) for 0 .. 20;
multi zeckendorf(0) { '0' } multi zeckendorf($n is copy) {
constant FIBS = (1,2, *+* ... *).cache;
[~] map {
$n -= $_ if my $digit = $n >= $_;
+$digit;
}, reverse FIBS ...^ * > $n;
}</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
REXX
specific to 20
<lang rexx> /* REXX ***************************************************************
- 11.10.2012 Walter Pachl
- /
fib='13 8 5 3 2 1' Do i=6 To 1 By -1 /* Prepare Fibonacci Numbers */
Parse Var fib f.i fib /* f.1 ... f.7 */ End
Do n=0 To 20 /* for all numbers in the task */
m=n /* copy of number */
r= /* result for n */
Do i=6 To 1 By -1 /* loop through numbers */
If m>=f.i Then Do /* f.i must be used */
r=r||1 /* 1 into result */
m=m-f.i /* subtract */
End
Else /* f.i is larger than the rest */
r=r||0 /* 0 into result */
End
r=strip(r,'L','0') /* strip leading zeros */
If r= Then r='0' /* take care of 0 */
Say right(n,2)': 'right(r,6) /* show result */
End</lang>
Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
generalized
This generalized REXX version will work for any Zeckendorf number (up to 100,000 decimal digits).
A list of Fibonacci numbers (in ascending order) is generated large enough to handle the Nth Zeckendorf number.
<lang rexx>/*REXX program calculates and displays the first N Zeckendorf numbers. */
numeric digits 100000 /*just in case user gets real ka─razy. */
parse arg N . /*let the user specify the upper limit.*/
if N== | N=="," then n=20; w= length(N) /*Not specified? Then use the default.*/
@.1= 1 /*start the array with 1 and 2. */
@.2= 2; do #=3 until #>=N; p= #-1; pp= #-2 /*build a list of Fibonacci numbers. */
@.#= @.p + @.pp /*sum the last two Fibonacci numbers. */
end /*#*/ /* [↑] #: contains a Fibonacci list.*/
do j=0 to N; parse var j x z /*task: process zero ──► N numbers.*/
do k=# by -1 for #; _= @.k /*process all the Fibonacci numbers. */
if x>=_ then do; z= z'1' /*is X>the next Fibonacci #? Append 1.*/
x= x - _ /*subtract this Fibonacci # from index.*/
end
else z= z'0' /*append zero (0) to the Fibonacci #. */
end /*k*/
say ' Zeckendorf' right(j, w) "=" right(z+0, 30) /*display a number.*/
end /*j*/ /*stick a fork in it, we're all done. */</lang>
- output when using the default input:
Zeckendorf 0 = 0
Zeckendorf 1 = 1
Zeckendorf 2 = 10
Zeckendorf 3 = 100
Zeckendorf 4 = 101
Zeckendorf 5 = 1000
Zeckendorf 6 = 1001
Zeckendorf 7 = 1010
Zeckendorf 8 = 10000
Zeckendorf 9 = 10001
Zeckendorf 10 = 10010
Zeckendorf 11 = 10100
Zeckendorf 12 = 10101
Zeckendorf 13 = 100000
Zeckendorf 14 = 100001
Zeckendorf 15 = 100010
Zeckendorf 16 = 100100
Zeckendorf 17 = 100101
Zeckendorf 18 = 101000
Zeckendorf 19 = 101001
Zeckendorf 20 = 101010
generic
This generic REXX version will generate up to the Nth Zeckendorf numbers (up to 100,000 decimal digits) by
using binary numbers that don't have two consecutive 11s within their binary version.
There isn't any need to generate a Fibonacci series with this method. This method is extremely fast. <lang REXX>/*REXX program calculates and displays the first N Zeckendorf numbers. */ numeric digits 100000 /*just in case user gets real ka─razy. */ parse arg N . /*let the user specify the upper limit.*/ if N== | N=="," then n=20; w= length(N) /*Not specified? Then use the default.*/ z=0 /*the index of a Zeckendorf number. */
do j=0 until z>N; _=x2b( d2x(j) ) /*task: process zero ──► N. */ if pos(11, _) \== 0 then iterate /*are there two consecutive ones (1s) ?*/ say ' Zeckendorf' right(z, w) "=" right(_+0, 30) /*display a number.*/ z= z + 1 /*bump the Zeckendorf number counter.*/ end /*j*/ /*stick a fork in it, we're all done. */</lang>
- output is identical to the previous (generalized) version.
Ring
<lang ring>
- Project : Zeckendorf number representation
see "0 0" + nl for n = 1 to 20
see "" + n + " " + zeckendorf(n) + nl
next
func zeckendorf(n)
fib = list(45)
fib[1] = 1
fib[2] = 1
i = 2
o = ""
while fib[i] <= n
i = i + 1
fib[i] = fib[i-1] + fib[i-2]
end
while i != 2
i = i - 1
if n >= fib[i]
o = o + "1"
n = n - fib[i]
else
o = o + "0"
ok
end
return o
</lang> Output:
0 0 1 1 2 10 3 100 4 101 5 1000 6 1001 7 1010 8 10000 9 10001 10 10010 11 10100 12 10101 13 100000 14 100001 15 100010 16 100100 17 100101 18 101000 19 101001 20 101010
Ruby
Featuring a method doubling as an enumerator. <lang ruby>def zeckendorf
return to_enum(__method__) unless block_given?
x = 0
loop do
bin = x.to_s(2)
yield bin unless bin.include?("11")
x += 1
end
end
zeckendorf.take(21).each_with_index{|x,i| puts "%3d: %8s"% [i, x]}</lang>
<lang ruby>def zeckendorf(n)
return 0 if n.zero?
fib = [1,2]
fib << fib[-2] + fib[-1] while fib[-1] < n
dig = ""
fib.reverse_each do |f|
if f <= n
dig, n = dig + "1", n - f
else
dig += "0"
end
end
dig.to_i
end
for i in 0..20
puts '%3d: %8d' % [i, zeckendorf(i)]
end</lang> As oneliner.
<lang ruby>def zeckendorf(n)
0.step.lazy.map { |x| x.to_s(2) }.reject { |z| z.include?("11") }.first(n)
end
zeckendorf(21).each_with_index{ |x,i| puts "%3d: %8s"% [i, x] } </lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Scala
<lang scala>def zNum( n:BigInt ) : String = {
if( n == 0 ) return "0" // Short-circuit this and return zero if we were given zero
val v = n.abs
val fibs : Stream[BigInt] = { def series(i:BigInt,j:BigInt):Stream[BigInt] = i #:: series(j, i+j); series(1,0).tail.tail.tail }
def z( v:BigInt ) : List[BigInt] = if(v == 0) List() else {val m = fibs(fibs.indexWhere(_>v) - 1); m :: z(v-m)}
val zv = z(v)
// Walk the list of fibonacci numbers from the number that matches the most significant down to 1,
// if the zeckendorf matchs then yield '1' otherwise '0'
val s = (for( i <- (fibs.indexWhere(_==zv(0)) to 0 by -1) ) yield {
if( zv.contains(fibs(i))) "1" else "0"
}).mkString if( n < 0 ) "-" + s // Using a negative-sign instead of twos-complement else s
}
// A little test...
(0 to 20) foreach( i => print( zNum(i) + "\n" ) )
</lang>
- Output:
0 1 10 100 101 1000 1001 1010 10000 10001 10010 10100 10101 100000 100001 100010 100100 100101 101000 101001 101010
Scheme
<lang scheme>(import (rnrs))
(define (getFibList maxNum n1 n2 fibs)
(if (> n2 maxNum)
fibs
(getFibList maxNum n2 (+ n1 n2) (cons n2 fibs))))
(define (getZeckendorf num)
(if (<= num 0)
"0"
(let ((fibs (getFibList num 1 2 (list 1))))
(getZeckString "" num fibs))))
(define (getZeckString zeck num fibs)
(let* ((curFib (car fibs))
(placeZeck (>= num curFib))
(outString (string-append zeck (if placeZeck "1" "0")))
(outNum (if placeZeck (- num curFib) num)))
(if (null? (cdr fibs))
outString
(getZeckString outString outNum (cdr fibs)))))
(let loop ((i 0))
(when (<= i 20)
(for-each
(lambda (item)
(display item))
(list "Z(" i "):\t" (getZeckendorf i)))
(newline)
(loop (+ i 1))))
</lang>
- Output:
Z(0): 0 Z(1): 1 Z(2): 10 Z(3): 100 Z(4): 101 Z(5): 1000 Z(6): 1001 Z(7): 1010 Z(8): 10000 Z(9): 10001 Z(10): 10010 Z(11): 10100 Z(12): 10101 Z(13): 100000 Z(14): 100001 Z(15): 100010 Z(16): 100100 Z(17): 100101 Z(18): 101000 Z(19): 101001 Z(20): 101010
Sidef
<lang ruby>func fib(n) is cached {
n < 2 ? 1
: (fib(n-1) + fib(n-2))
}
func zeckendorf(n) {
n == 0 && return '0'
var i = 1
++i while (fib(i) <= n)
gather {
while (--i > 0) {
var f = fib(i)
f > n ? (take '0')
: (take '1'; n -= f)
}
}.join
}
for n (0..20) {
printf("%4d: %8s\n", n, zeckendorf(n))
}</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Simula
<lang simula>BEGIN
INTEGER N, F0, F1, F2, D;
N := 20;
COMMENT CALCULATE D FROM ANY GIVEN N ;
F1 := 1; F2 := 2; F0 := F1 + F2; D := 2;
WHILE F0 < N DO BEGIN
F1 := F2; F2 := F0; F0 := F1 + F2; D := D + 1;
END;
BEGIN
COMMENT Sinclair ZX81 BASIC Solution ;
TEXT Z1, S1;
INTEGER I, J, Z;
INTEGER ARRAY F(1:D); ! 10 dim f(6) ;
F(1) := 1; ! 20 let f(1)=1 ;
F(2) := 2; ! 30 let f(2)=2 ;
FOR I := 3 STEP 1 UNTIL D DO BEGIN ! 40 for i=3 to 6 ;
F(I) := F(I-2) + F(I-1); ! 50 let f(i)=f(i-2)+f(i-1) ;
END; ! 60 next i ;
FOR I := 0 STEP 1 UNTIL N DO BEGIN ! 70 for i=0 to 20 ;
Z1 :- ""; ! 80 let z$="" ;
S1 :- " "; ! 90 let s$=" " ;
Z := I; ! 100 let z=i ;
FOR J := D STEP -1 UNTIL 1 DO BEGIN ! 110 for j=6 to 1 step -1 ;
IF J=1 THEN S1 :- "0"; ! 120 if j=1 then let s$="0" ;
IF NOT (Z<F(J)) THEN BEGIN ! 130 if z<f(j) then goto 180 ;
Z1 :- Z1 & "1"; ! 140 let z$=z$+"1" ;
Z := Z-F(J); ! 150 let z=z-f(j) ;
S1 :- "0"; ! 160 let s$="0" ;
END ELSE ! 170 goto 190 ;
Z1 :- Z1 & S1; ! 180 let z$=z$+s$ ;
END; ! 190 next j ;
OUTINT(I, 0); OUTCHAR(' '); ! 200 print i ; !" "; !;
IF I<10 THEN OUTCHAR(' '); ! 210 if i<10 then print " "; !;
OUTTEXT(Z1); OUTIMAGE; ! 220 print z$ ;
END; ! 230 next i ;
END;
END</lang>
- Output:
0 0 1 1 2 10 3 100 4 101 5 1000 6 1001 7 1010 8 10000 9 10001 10 10010 11 10100 12 10101 13 100000 14 100001 15 100010 16 100100 17 100101 18 101000 19 101001 20 101010
Sinclair ZX81 BASIC
Works on the 1k RAM model, albeit without much room for manoeuvre. (You'd like the Zeckendorf numbers further over towards the right-hand side of the screen? Sorry, can't spare the video RAM.) If you have 2k or more, you can replace the constant 6 with some higher value wherever it occurs in the program and enable yourself to represent bigger numbers in Zeckendorf form. <lang basic> 10 DIM F(6)
20 LET F(1)=1 30 LET F(2)=2 40 FOR I=3 TO 6 50 LET F(I)=F(I-2)+F(I-1) 60 NEXT I 70 FOR I=0 TO 20 80 LET Z$="" 90 LET S$=" "
100 LET Z=I 110 FOR J=6 TO 1 STEP -1 120 IF J=1 THEN LET S$="0" 130 IF Z<F(J) THEN GOTO 180 140 LET Z$=Z$+"1" 150 LET Z=Z-F(J) 160 LET S$="0" 170 GOTO 190 180 LET Z$=Z$+S$ 190 NEXT J 200 PRINT I;" "; 210 IF I<10 THEN PRINT " "; 220 PRINT Z$ 230 NEXT I</lang>
- Output:
0 0 1 1 2 10 3 100 4 101 5 1000 6 1001 7 1010 8 10000 9 10001 10 10010 11 10100 12 10101 13 100000 14 100001 15 100010 16 100100 17 100101 18 101000 19 101001 20 101010
Standard ML
<lang Standard ML> val zeckList = fn from => fn to =>
let open IntInf
val rec npow = fn n => fn 0 => fromInt 1 | m => n* (npow n (m-1)) ;
val fib = fn 0 => 1 | 1 => 1 | n => let val rec fb = fn x => fn y => fn 1=>y | n=> fb y (x+y) (n-1) in
fb 0 1 n
end;
val argminfi = fn n => (* lowest k with fibonacci number over n *)
let
val rec afb = fn k => if fib k > n then k else afb (k+1)
in
afb 0
end;
val Zeck = fn n =>
let
val rec calzk = fn (0,z) => (0,z)
| (n,z) => let val k = argminfi n in
calzk ( n - fib (k-1) , z + (npow 10 (k-3) ) )
end
in
#2 (calzk (n,0))
end
in
List.tabulate (toInt ( to - from) ,
fn i:Int.int => ( from + (fromInt i),
Zeck ( from + (fromInt i) ))) end; </lang> output <lang Standard ML> List.app ( fn e => print ( (IntInf.toString (#1 e)) ^" : "^ (IntInf.toString (#2 e)) ^ "\n" )) (zeckList 1 21) ; 1 : 1 2 : 10 3 : 100 4 : 101 5 : 1000 6 : 1001 7 : 1010 8 : 10000 9 : 10001 10 : 10010 11 : 10100 12 : 10101 13 : 100000 14 : 100001 15 : 100010 16 : 100100 17 : 100101 18 : 101000 19 : 101001 20 : 101010
zeckList 0x21e320a3 0x21e320a4 ; val it = [(568533155, 100100100101001001001000000100100010100101)]:
: (IntInf.int * IntInf.int) list
</lang>
Tcl
<lang tcl>package require Tcl 8.5
- Generates the Fibonacci sequence (starting at 1) up to the largest item that
- is no larger than the target value. Could use tricks to precompute, but this
- is actually a pretty cheap linear operation.
proc fibseq target {
set seq {}; set prev 1; set fib 1
for {set n 1;set i 1} {$fib <= $target} {incr n} {
for {} {$i < $n} {incr i} { lassign [list $fib [incr fib $prev]] prev fib } if {$fib <= $target} { lappend seq $fib }
} return $seq
}
- Produce the given Zeckendorf number.
proc zeckendorf n {
# Special case: only value that begins with 0
if {$n == 0} {return 0}
set zs {}
foreach f [lreverse [fibseq $n]] {
lappend zs [set z [expr {$f <= $n}]] if {$z} {incr n [expr {-$f}]}
} return [join $zs ""]
}</lang> Demonstration <lang tcl>for {set i 0} {$i <= 20} {incr i} {
puts [format "%2d:%9s" $i [zeckendorf $i]]
}</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
uBasic/4tH
<lang>For x = 0 to 20 ' Print Zeckendorf numbers 0 - 20
Print x, Push x : Gosub _Zeckendorf ' get Zeckendorf number repres. Print ' terminate line
Next
End
_Fibonacci
Push Tos() ' duplicate TOS()
@(0) = 0 ' This function returns the
@(1) = 1 ' Fibonacci number which is smaller
' or equal to TOS()
Do While @(1) < Tos() + 1
Push (@(1))
@(1) = @(0) + @(1) ' get next Fibonacci number
@(0) = Pop()
Loop ' loop if not exceeded TOS()
Gosub _Drop ' clear TOS() Push @(0) ' return Fibonacci number
Return
_Zeckendorf
GoSub _Fibonacci ' This function breaks TOS() up
Print Tos(); ' into its Zeckendorf components
Push -(Pop() - Pop()) ' first digit is always there
' the remainder to resolve
Do While Tos() ' now go for the next digits
GoSub _Fibonacci
Print " + ";Tos(); ' print the next digit
Push -(Pop() - Pop())
Loop
Gosub _Drop ' clear TOS()
Return ' and return
_Drop
If Pop()%1 = 0 Then Return ' This function clears TOS()</lang>
Output:
0 0 1 1 2 2 3 3 4 3 + 1 5 5 6 5 + 1 7 5 + 2 8 8 9 8 + 1 10 8 + 2 11 8 + 3 12 8 + 3 + 1 13 13 14 13 + 1 15 13 + 2 16 13 + 3 17 13 + 3 + 1 18 13 + 5 19 13 + 5 + 1 20 13 + 5 + 2 0 OK, 0:901
VBA
<lang vb>Private Function zeckendorf(ByVal n As Integer) As Integer
Dim r As Integer: r = 0
Dim c As Integer
Dim fib As New Collection
fib.Add 1
fib.Add 1
Do While fib(fib.Count) < n
fib.Add fib(fib.Count - 1) + fib(fib.Count)
Loop
For i = fib.Count To 2 Step -1
c = n >= fib(i)
r = r + r - c
n = n + c * fib(i)
Next i
zeckendorf = r
End Function
Public Sub main()
Dim i As Integer
For i = 0 To 20
Debug.Print Format(i, "@@"); ":"; Format(WorksheetFunction.Dec2Bin(zeckendorf(i)), "@@@@@@@")
Next i
End Sub</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
VBScript
<lang vb> Function Zeckendorf(n) num = n Set fibonacci = CreateObject("System.Collections.Arraylist") fibonacci.Add 1 : fibonacci.Add 2 i = 1 Do While fibonacci(i) < num fibonacci.Add fibonacci(i) + fibonacci(i-1) i = i + 1 Loop tmp = "" For j = fibonacci.Count-1 To 0 Step -1 If fibonacci(j) <= num And (tmp = "" Or Left(tmp,1) <> "1") Then tmp = tmp & "1" num = num - fibonacci(j) Else tmp = tmp & "0" End If Next Zeckendorf = CLng(tmp) End Function
'testing the function For k = 0 To 20 WScript.StdOut.WriteLine k & ": " & Zeckendorf(k) Next </lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Wren
<lang ecmascript>import "/fmt" for Fmt
var LIMIT = 46 // to stay within range of signed 32 bit integer
var fibonacci = Fn.new { |n|
if (n < 2 || n > LIMIT) Fiber.abort("n must be between 2 and %(LIMIT)")
var fibs = List.filled(n, 1)
for (i in 2...n) fibs[i] = fibs[i - 1] + fibs[i - 2]
return fibs
}
var fibs = fibonacci.call(LIMIT)
var zeckendorf = Fn.new { |n|
if (n < 0) Fiber.abort("n must be non-negative")
if (n < 2) return n.toString
var lastFibIndex = 1
for (i in 2..LIMIT) {
if (fibs[i] > n) {
lastFibIndex = i - 1
break
}
}
n = n - fibs[lastFibIndex]
lastFibIndex = lastFibIndex - 1
var zr = "1"
for (i in lastFibIndex..1) {
if (fibs[i] <= n) {
zr = zr + "1"
n = n - fibs[i]
} else {
zr = zr + "0"
}
}
return zr
}
System.print(" n z") for (i in 0..20) Fmt.print("$2d : $s", i, zeckendorf.call(i))</lang>
- Output:
n z 0 : 0 1 : 1 2 : 10 3 : 100 4 : 101 5 : 1000 6 : 1001 7 : 1010 8 : 10000 9 : 10001 10 : 10010 11 : 10100 12 : 10101 13 : 100000 14 : 100001 15 : 100010 16 : 100100 17 : 100101 18 : 101000 19 : 101001 20 : 101010
XPL0
<lang XPL0>include c:\cxpl\codes; \intrinsic 'code' declarations
proc Zeckendorf(N); \Display Zeckendorf number (N <= 20) int N; int Fib, LZ, I; [Fib:= [1, 2, 3, 5, 8, 13]; \Fibonacci sequence LZ:= true; \suppress leading zeros for I:= 5 downto 1 do
[if N >= Fib(I) then [N:= N-Fib(I); ChOut(0, ^1); LZ:= false]
else ChOut(0, if LZ then ^ else ^0);
];
ChOut(0, N+^0); \output final digit, which can be 0 ];
int N; [for N:= 0 to 20 do
[if N<10 then ChOut(0,^ ); IntOut(0, N); Text(0, ": ");
Zeckendorf(N); CrLf(0);
];
]</lang>
Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
Yabasic
<lang Yabasic>sub Zeckendorf(n) local i, n$, c
do n$ = bin$(i) if not instr(n$,"11") then print c,":\t",n$ if c = n break c = c + 1 end if i = i + 1 loop end sub
Zeckendorf(20) </lang>
zkl
<lang zkl> // return powers (0|1) of fib sequence (1,2,3,5,8...) that sum to n fcn zeckendorf(n){ //-->String of 1s & 0s, no consecutive 1's
if(n<=0) return("0");
fibs:=fcn(ab){ ab.append(ab.sum()).pop(0) }.fp(L(1,2));
(0).pump(*,List,fibs,'wrap(fib){ if(fib>n)Void.Stop else fib })
.reverse()
.pump(String,fcn(fib,rn){
if(fib>rn.value)"0" else { rn.set(rn.value-fib); "1" } }.fp1(Ref(n)))
}</lang> <lang zkl>[0..20].pump(Console.println,fcn(n){ "%2d: %8s".fmt(n,zeckendorf(n)) });</lang>
- Output:
0: 0 1: 1 2: 10 3: 100 4: 101 5: 1000 6: 1001 7: 1010 8: 10000 9: 10001 10: 10010 11: 10100 12: 10101 13: 100000 14: 100001 15: 100010 16: 100100 17: 100101 18: 101000 19: 101001 20: 101010
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