Calkin-Wilf sequence: Difference between revisions
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* Find the position of the number <big>'''<sup>83116</sup>'''<big>'''/'''</big>'''<sub>51639</sub>'''</big> in the Calkin-Wilf sequence.
;Related tasks:
:* [[Fusc sequence]].
;See also:
Line 35 ⟶ 37:
* [[Continued fraction/Arithmetic/Construct from rational number]]
<br><br>
=={{header|11l}}==
{{trans|Nim}}
<syntaxhighlight lang="11l">T CalkinWilf
n = 1
d = 1
F ()()
V r = (.n, .d)
.n = 2 * (.n I/ .d) * .d + .d - .n
swap(&.n, &.d)
R r
print(‘The first 20 terms of the Calkwin-Wilf sequence are:’)
V cw = CalkinWilf()
[String] seq
L 20
V (n, d) = cw()
seq.append(I d == 1 {String(n)} E n‘/’d)
print(seq.join(‘, ’))
cw = CalkinWilf()
V index = 1
L cw() != (83116, 51639)
index++
print("\nThe element 83116/51639 is at position "index‘ in the sequence.’)</syntaxhighlight>
{{out}}
<pre>
The first 20 terms of the Calkwin-Wilf sequence are:
1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4, 1/5, 5/4, 4/7, 7/3, 3/8
The element 83116/51639 is at position 123456789 in the sequence.
</pre>
=={{header|ALGOL 68}}==
Uses code from the [[Arithmetic/Rational]] and [[Continued fraction/Arithmetic/Construct from rational number]] tasks.
<syntaxhighlight lang="algol68">BEGIN
# Show elements 1-20 of the Calkin-Wilf sequence as rational numbers #
# also show the position of a specific element in the seuence #
# Uses code from the Arithmetic/Rational #
# & Continued fraction/Arithmetic/Construct from rational number tasks #
# Code from the Arithmetic/Rational task #
# ============================================================== #
MODE FRAC = STRUCT( INT num #erator#, den #ominator#);
PROC gcd = (INT a, b) INT: # greatest common divisor #
(a = 0 | b |: b = 0 | a |: ABS a > ABS b | gcd(b, a MOD b) | gcd(a, b MOD a));
PROC lcm = (INT a, b)INT: # least common multiple #
a OVER gcd(a, b) * b;
PRIO // = 9; # higher then the ** operator #
OP // = (INT num, den)FRAC: ( # initialise and normalise #
INT common = gcd(num, den);
IF den < 0 THEN
( -num OVER common, -den OVER common)
ELSE
( num OVER common, den OVER common)
FI
);
OP + = (FRAC a, b)FRAC: (
INT common = lcm(den OF a, den OF b);
FRAC result := ( common OVER den OF a * num OF a + common OVER den OF b * num OF b, common );
num OF result//den OF result
);
OP - = (FRAC a, b)FRAC: a + -b,
* = (FRAC a, b)FRAC: (
INT num = num OF a * num OF b,
den = den OF a * den OF b;
INT common = gcd(num, den);
(num OVER common) // (den OVER common)
);
OP - = (FRAC frac)FRAC: (-num OF frac, den OF frac);
# ============================================================== #
# end code from the Arithmetic/Rational task #
# code from the Continued fraction/Arithmetic/Construct from rational number task #
# ================================================================================#
# returns the quotient of numerator over denominator and sets #
# numerator and denominator to the next values for #
# the continued fraction #
PROC r2cf = ( REF INT numerator, REF INT denominator )INT:
IF denominator = 0
THEN 0
ELSE INT quotient := numerator OVER denominator;
INT prev numerator = numerator;
numerator := denominator;
denominator := prev numerator MOD denominator;
quotient
FI # r2cf # ;
# ====================================================================================#
# end code from the Continued fraction/Arithmetic/Construct from rational number task #
# Additional FRACrelated operators #
OP * = ( INT a, FRAC b )FRAC: ( num OF b * a ) // den OF b;
OP / = ( FRAC a, b )FRAC: ( num OF a * den OF b ) // ( num OF b * den OF a );
OP FLOOR = ( FRAC a )INT: num OF a OVER den OF a;
OP + = ( INT a, FRAC b )FRAC: ( a // 1 ) + b;
FRAC one = 1 // 1;
# returns the first n elements of the Calkin-Wilf sequence #
PROC calkin wilf = ( INT n )[]FRAC:
BEGIN
[ 1 : n ]FRAC q;
IF n > 0 THEN
q[ 1 ] := 1 // 1;
FOR i FROM 2 TO UPB q DO
q[ i ] := one / ( ( 2 * FLOOR q[ i - 1 ] ) + one - q[ i - 1 ] )
OD
FI;
q
END # calkin wilf # ;
# returns the position of a FRAC in the Calkin-Wilf sequence by computing its #
# continued fraction representation and converting that to a bit string #
# - the position must fit in a 2-bit number #
PROC position in calkin wilf sequence = ( FRAC f )INT:
IF INT result := 0;
[ 1 : 32 ]INT cf; FOR i FROM LWB cf TO UPB cf DO cf[ i ] := 0 OD;
INT num := num OF f;
INT den := den OF f;
INT cf length := 0;
FOR i FROM LWB cf WHILE den /= 0 DO
cf[ cf length := i ] := r2cf( num, den )
OD;
NOT ODD cf length
THEN # the continued fraction does not have an odd length #
-1
ELSE # the continued fraction has an odd length so we can compute the seuence length #
# build the number by alternating d 1s and 0s where d is the digits of the #
# continued fraction, starting at the least significant #
INT digit := 1;
FOR d pos FROM cf length BY -1 TO 1 DO
FOR i TO cf[ d pos ] DO
result *:= 2 +:= digit
OD;
digit := IF digit = 0 THEN 1 ELSE 0 FI
OD;
result
FI # position in calkin wilf sequence # ;
BEGIN # task #
# get and show the first 20 Calkin-Wilf sequence numbers #
[]FRAC cw = calkin wilf( 20 );
print( ( "The first 20 elements of the Calkin-Wilf sequence are:", newline, " " ) );
FOR n FROM LWB cw TO UPB cw DO
FRAC sn = cw[ n ];
print( ( " ", whole( num OF sn, 0 ), "/", whole( den OF sn, 0 ) ) )
OD;
print( ( newline ) );
# show the position of a specific element in the sequence #
print( ( "Position of 83116/51639 in the sequence: "
, whole( position in calkin wilf sequence( 83116//51639 ), 0 )
)
)
END
END</syntaxhighlight>
{{out}}
<pre>
The first 20 elements of the Calkin-Wilf sequence are:
1/1 1/2 2/1 1/3 3/2 2/3 3/1 1/4 4/3 3/5 5/2 2/5 5/3 3/4 4/1 1/5 5/4 4/7 7/3 3/8
Position of 83116/51639 in the sequence: 123456789
</pre>
=={{header|AppleScript}}==
<
on CalkinWilfSequence(n)
script o
Line 134 ⟶ 308:
set AppleScript's text item delimiters to astid
set output to output & (linefeed & "83116/51639 is term number " & positionResult)
return output</
{{output}}
<
1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4/1, 1/5, 5/4, 4/7, 7/3, 3/8
Ditto using binary run-length encoding:
1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4/1, 1/5, 5/4, 4/7, 7/3, 3/8
83116/51639 is term number 123456789"</
=={{header|Arturo}}==
<
d: new 1
calkinWilf: function [] .export:[n,d] [
Line 171 ⟶ 344:
print ""
print ["The element" ~"|target\0|/|target\1|" "is at position" indx "in the sequence."]</
{{out}}
Line 179 ⟶ 352:
The element 83116/51639 is at position 123456789 in the sequence.</pre>
=={{header|BQN}}==
BQN does not have rational number arithmetic yet, so it is manually implemented.
Part 2 runs in ~150 secs on [[CBQN]].
<code>GCD</code> and <code>_while_</code> are idioms from [https://mlochbaum.github.io/bqncrate/ BQNcrate].
<syntaxhighlight lang="bqn">GCD ← {m 𝕊⍟(0<m←𝕨|𝕩) 𝕨}
_while_ ← {𝔽⍟𝔾∘𝔽_𝕣_𝔾∘𝔽⍟𝔾𝕩}
Sim ← { # Simplify a fraction
x𝕊1: 𝕨‿1;
0𝕊y: 0‿𝕩;
⌊𝕨‿𝕩 ÷ 𝕨 GCD 𝕩
}
Add ← { # Add two fractions
0‿b 𝕊 𝕩: 𝕩;
𝕨 𝕊 0‿y: 𝕨;
a‿b 𝕊 x‿y:
((a×y)+x×b) Sim b×y
}
Next ← {n‿d: ⌽(2×⌊÷´n‿d)‿1 Add (d-n)‿d} # Next term
Cal ← {Next⍟𝕩 1‿1}
•Show Cal 1+↕20
•Show {
cnt‿fr:
⟨cnt+1,Next fr⟩
} _while_ {
cnt‿fr:
fr ≢ 83116‿51639
} ⟨1,1‿1⟩</syntaxhighlight>
<syntaxhighlight lang="bqn">⟨ ⟨ 1 2 ⟩ ⟨ 2 1 ⟩ ⟨ 1 3 ⟩ ⟨ 3 2 ⟩ ⟨ 2 3 ⟩ ⟨ 3 1 ⟩ ⟨ 1 4 ⟩ ⟨ 4 3 ⟩ ⟨ 3 5 ⟩ ⟨ 5 2 ⟩ ⟨ 2 5 ⟩ ⟨ 5 3 ⟩ ⟨ 3 4 ⟩ ⟨ 4 1 ⟩ ⟨ 1 5 ⟩ ⟨ 5 4 ⟩ ⟨ 4 7 ⟩ ⟨ 7 3 ⟩ ⟨ 3 8 ⟩ ⟨ 8 5 ⟩ ⟩
⟨ 123456789 ⟨ 83116 51639 ⟩ ⟩</syntaxhighlight>
You can try Part 1 [https://mlochbaum.github.io/BQN/try.html#code=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 here.] Second part can and will hang your browser, so it is best to try locally on [[CBQN]].
=={{header|Bracmat}}==
{{trans|Python}}
<syntaxhighlight lang="bracmat">( 1:?a
& 0:?i
& whl
' ( 1+!i:<20:?i
& (2*div$(!a,1)+1+-1*!a)^-1:?a
& out$!a
)
& ( r2cf
= floor
. div$(!arg,1):?floor
& ( !floor:!arg
| !floor r2cf$((!arg+-1*!floor)^-1)
)
)
& ( get-term-num
= ans dig pwr
. (0,1,1):(?ans,?dig,?pwr)
& r2cf$!arg:?n
& map
$ ( (
=
. whl
' ( !arg+-1:~<0:?arg
& !dig*!pwr+!ans:?ans
& 2*!pwr:?pwr
)
& 1+-1*!dig:?dig
)
. !n
)
& !ans
)
& out$(get-term-num$83116/51639)
);</syntaxhighlight>
{{out}}
<pre>1/2
2
1/3
3/2
2/3
3
1/4
4/3
3/5
5/2
2/5
5/3
3/4
4
1/5
5/4
4/7
7/3
3/8
123456789</pre>
=={{header|C++}}==
{{libheader|Boost}}
<
#include <vector>
#include <boost/rational.hpp>
Line 234 ⟶ 499:
rational r(83116, 51639);
std::cout << r << " is the " << term_number(r) << "th term of the sequence.\n";
}</
{{out}}
Line 260 ⟶ 525:
20: 3/8
83116/51639 is the 123456789th term of the sequence.
</pre>
=={{header|EasyLang}}==
{{trans|Nim}}
<syntaxhighlight>
subr first
n = 1 ; d = 1
.
proc next . .
n = 2 * (n div d) * d + d - n
swap n d
.
print "The first 20 terms of the Calkwin-Wilf sequence are:"
first
for i to 20
write n & "/" & d & " "
next
.
print ""
#
first
i = 1
while n <> 83116 or d <> 51639
next
i += 1
.
print "83116/51639 is at position " & i
</syntaxhighlight>
{{out}}
<pre>
The first 20 terms of the Calkwin-Wilf sequence are:
1/1 1/2 2/1 1/3 3/2 2/3 3/1 1/4 4/3 3/5 5/2 2/5 5/3 3/4 4/1 1/5 5/4 4/7 7/3 3/8
83116/51639 is at position 123456789
</pre>
=={{header|EDSAC order code}}==
===Find first n terms===
{{trans|Pascal}}
<syntaxhighlight lang="edsac">
[For Rosetta Code. EDSAC program, Initial Orders 2.
Prints the first 20 terms of the Calkin-Wilf sequence.
Uses term{n} to calculate term{n + 1}.]
[Print subroutine for non-negative 17-bit integers.
Parameters: 0F = integer to be printed (not preserved)
1F = character for leading zero (preserved)
Workspace: 4F, 5F. Even address; 40 locations]
T 56 K [define load address]
GKA3FT34@A1FT35@S37@T36@AFT5FT4FH38#@V4DH30@
R32FR16FYFE23@O35@A2FT36@T5FV4DYFL8FT4DA5FL1024F
UFA36@G16@OFTFT35@A36@G17@ZFPFPFP4FT1714FZ219D
[Main routine]
T 100 K [define load address]
G K [set up relative addressing via @ (theta)]
[Constants]
[0] P 10 F [maximum index = 20, edit ad lib.]
[1] P D [constant 1]
[Teleprinter characters]
[2] # F [set figures mode]
[3] C F [colon (in figures mode)]
[4] X F [slash (in figures mode)]
[5] ! F [space]
[6] @ F [carriage return]
[7] & F [line feed]
[8] K 4096 F [null]
[Variables]
[9] P F [index]
[10] P F [a (where term = a/b)]
[11] P F [b]
[Enter with acc = 0]
[12] O 2 @ [set teleprinter to figures]
A 1 @ [acc := 1]
U 9 @ [index := 1]
U 10 @ [a := 1]
T 11 @ [b := 1 (and clear acc)]
E 34 @ [jump to print first term]
[Loop back here if not yet printed enough terms]
[18] A @ [restore index after test]
A 1 @ [add 1]
T 9 @ [update index]
[Calculate next term. New b := a + b - 2(a mod b).
Code below calculates c := (a mod b) - b, then new b := a - b - 2*c]
A 10 @ [acc := a]
[22] S 11 @ [subtract b]
E 22 @ [if acc >= 0, subtract again]
T F [result c < 0, store in 0F]
A 10 @ [acc := a]
S 11 @ [subtract b]
S F [subtract c]
S F [subtract c]
T F [new b = a - b - 2*c; store in 0F]
A 11 @ [acc := old b]
T 10 @ [copy to a]
A F [acc := new b]
T 11 @ [copy to b]
[Print index and a/b. Assume acc = 0 here.]
[34] A 5 @ [space to replace leading 0's]
T 1 F [pass to print subroutine]
A 9 @ [acc := index]
T F [pass to print subroutine]
[38] A 38 @ [for return from subroutine]
G 56 F [call subroutine, clears acc]
O 3 @ [print colon]
O 5 @ [print space]
A 8 @ [null to replace leading 0's]
T 1 F [pass to print subroutine]
A10@ TF A46@ G56F O4@ [print a followed by slash]
A11@ TF A51@ G56F O6@ O7@ [print b followed by CR LF]
[Test whether enough terms have been printed]
A 9 @ [acc := index]
S @ [subtract maximum index]
G 18 @ [loop back if acc < 0]
[Exit]
O 8 @ [print null to flush teleprinter buffer]
Z F [stop]
E 12 Z [relative address of entry point]
P F [enter with acc = 0]
[end]
</syntaxhighlight>
{{out}}
<pre>
1: 1/1
2: 1/2
3: 2/1
4: 1/3
5: 3/2
6: 2/3
7: 3/1
8: 1/4
9: 4/3
10: 3/5
11: 5/2
12: 2/5
13: 5/3
14: 3/4
15: 4/1
16: 1/5
17: 5/4
18: 4/7
19: 7/3
20: 3/8
</pre>
===Find index of a given term===
{{trans|Pascal}}
<syntaxhighlight lang="edsac">
[For Rosetta Code. EDSAC program, Initial Orders 2.]
[Finds the index of a given rational in the Calkin-Wilf series.]
[Library subroutine R2: input of positive integers.
Runs during input of the program, and is then overwritten.
Allows integers to be written in decimal, rather than as "pseudo-orders".
See Wilkes, Williams & Gill, 1951 edn, pp. 96-97, 148.]
T 54 K [to access integers via C parameter]
P 110 F [where to load integers]
GKT20FVDL8FA40DUDTFI40FA40FS39FG@S2FG23FA5@T5@E4@E13Z
T #C [tell R2 where to load integers]
[F after each integer except the last, and # after the last.]
83116F51639#
[Modified library subroutine P7.
Prints signed integer up to 10 digits, left-justified.
Input: Number to be printed is at 0D.
54 locations. Load at even address. Workspace 4D.]
T 56 K
GKA3FT42@A49@T31@ADE10@T31@A48@T31@SDTDH44#@NDYFLDT4DS43@
TFH17@S17@A43@G23@UFS43@T1FV4DAFG50@SFLDUFXFOFFFSFL4FT4DA49@
T31@A1FA43@G20@XFP1024FP610D@524D!FO46@O26@XFSFL8FT4DE39@
[Main routine.]
T 120 K [define load address (must be even)]
G K [set up relative addressing via @ (theta)]
[Put 35-bit values first, to ensure each is at an even address]
[Variables]
[0] P F P F [a]
[2] P F P F [b]
[4] P F P F [power of 2]
[6] P F P F [calculated index]
[Constants]
T8#Z PF T8Z [clears sandwich digit between 8 and 9]
[8] P D P F [35-bit constant 1]
[Teleprinter characters]
[10] # F [set figures mode]
[11] X F [slash (in figures mode)]
[12] K 2048 F [set letters mode]
[13] I F [letter I]
[14] R F [letter R]
[15] ! F [space]
[16] @ F [carriage return]
[17] & F [line feed]
[18] K 4096 F [null char]
[Enter with acc = 0]
[19] A #C [acc := initial a]
T #@ [copy to variable]
A 2#C [acc := initial b]
T 2#@ [copy to variable]
[23] A 8#@ [acc := 1]
[24] T 4#@ [initialize power of 2]
T 6#@ [initialize index to 0]
[Loop]
[26] A #@ [acc := a]
[27] S 2#@ [subtract b]
[28] E 33 @ [jump if a >= b]
[Here if a < b]
T D [store a - b in 0D]
S D [negate]
T 2#@ [b := b - a]
E 40 @ [join common code]
[Here if a >= b]
[33] S 8#@ [acc = a - b; test for a = b]
G 45 @ [jump out of loop if so]
A 8#@ [restore a - b]
T #@ [a := a - b]
A 6#@ [acc := index]
A 4#@ [inc index by power of 2]
T 6#@
[Code common to both cases]
[40] A 4#@ [acc := power of 2]
L D [shift left]
G 76 @
T 4#@ [update power of 2]
E 26 @ [loop back]
[Exit from loop.]
[45] T D [dump acc to clear it]
A 6#@ [acc := index]
A 4#@ [add power of 2 ]
T 6#@ [store final value of index]
[Finished calcualting index, now do printing]
O 10 @ [set teleprinter to figures]
A #C [acc := initial a]
T D [to 0D for printing]
[52] A 52 @ [for return from subroutine]
G 56 F [call print subroutine, clears acc]
O 11 @ [print slash]
A 2#C [print initial b similarly]
T D
[57] A 57 @
G 56 F
O 12 @ [set teleprinter to letters and print ' IS AT ']
O15@ O13@ O27@ O15@ O23@ O24@ O15@
O 10 @ [set teleprinter to figures]
A 6#@ [acc := calculated index]
T D [send to print subroutine]
[70] A 70 @
G 56 F
[72] O16@ O17@ [print CR, LF]
O 18 @ [print null to flush teleprinter buffer]
Z F [stop]
[Here if power of 2 goes negative (accumulator overflow)]
[76] O 12 @ [set teleprinter to letters]
O28@ O14@ O14@ O76@ O14@ [print'ERROR']
G 72 @ [jump to common exit]
E 19 Z [relative address of entry point]
P F [enter with acc = 0]
</syntaxhighlight>
{{out}}
<pre>
83116/51639 IS AT 123456789
</pre>
=={{header|F_Sharp|F#}}==
===The Function===
<
// Calkin Wilf Sequence. Nigel Galloway: January 9th., 2021
let cW=Seq.unfold(fun(n)->Some(n,seq{for n,g in n do yield (n,n+g); yield (n+g,g)}))(seq[(1,1)])|>Seq.concat
</syntaxhighlight>
===The Tasks===
; first 20
<
cW |> Seq.take 20 |> Seq.iter(fun(n,g)->printf "%d/%d " n g);printfn ""
</syntaxhighlight>
{{out}}
<pre>
Line 278 ⟶ 803:
</pre>
; Indexof 83116/51639
<
printfn "%d" (1+Seq.findIndex(fun n->n=(83116,51639)) cW)
</syntaxhighlight>
{{out}}
<pre>
Line 287 ⟶ 812:
=={{header|Factor}}==
{{works with|Factor|0.99 2020-08-14}}
<
math.continued-fractions math.functions math.parser prettyprint
sequences strings vectors ;
Line 307 ⟶ 832:
20 calkin-wilf ltake [ pprint bl ] leach nl nl
83116/51639 cw-index "83116/51639 is at index %d.\n" printf</
{{out}}
<pre>
Line 315 ⟶ 840:
83116/51639 is at index 123456789.
</pre>
=={{header|Forth}}==
{{works with|gforth|0.7.3}}
<syntaxhighlight lang="forth">\ Calkin-Wilf sequence
: frac. swap . ." / " . ;
: cw-next ( num den -- num den ) 2dup / over * 2* over + rot - ;
: cw-seq ( n -- )
1 1 rot
0 do
cr 2dup frac. cw-next
loop 2drop ;
variable index
variable bit-state
variable bit-position
: r2cf-next ( num1 den1 -- num2 den2 u ) swap over >r s>d r> sm/rem ;
: n2bitlength ( n -- )
bit-state @ if
1 swap lshift 1- bit-position @ lshift index +!
else drop then ;
: index-init true bit-state ! 0 bit-position ! 0 index ! ;
: index-build ( n -- )
dup n2bitlength bit-position +! bit-state @ invert bit-state ! ;
: index-finish ( n 0 -- ) 2drop -1 bit-position +! 1 index-build ;
: cw-index ( num den -- )
index-init
begin r2cf-next index-build dup 0<> while repeat
index-finish ;
: cw-demo
20 cw-seq
cr 83116 51639 2dup frac. cw-index index @ . ;
cw-demo</syntaxhighlight>
{{out}}
<pre>1 / 1
1 / 2
2 / 1
1 / 3
3 / 2
2 / 3
3 / 1
1 / 4
4 / 3
3 / 5
5 / 2
2 / 5
5 / 3
3 / 4
4 / 1
1 / 5
5 / 4
4 / 7
7 / 3
3 / 8
83116 / 51639 123456789 ok</pre>
=={{header|FreeBASIC}}==
Uses the code from [[Greatest common divisor#FreeBASIC]] as an include.
<
type rational
Line 427 ⟶ 1,012:
q.num = 83116
q.den = 51639
print disp_rational(q)+" is the "+str(frac_to_int(q))+"th term."</
{{out}}
Line 452 ⟶ 1,037:
20 3/8
83116/51639 is the 123456789th term.</pre>
=={{header|Fōrmulæ}}==
{{FormulaeEntry|page=https://formulae.org/?script=examples/Calkin-Wilf_correspondence}}
=={{header|Go}}==
{{trans|Wren}}
Go just has arbitrary precision rational numbers which we use here whilst assuming the numbers needed for this task can be represented exactly by the 64 bit built-in types.
<
import (
Line 554 ⟶ 1,134:
tn := getTermNumber(cf)
fmt.Printf("%s is the %sth term of the sequence.\n", r.RatString(), commatize(tn))
}</
{{out}}
Line 582 ⟶ 1,162:
83116/51639 is the 123,456,789th term of the sequence.
</pre>
=={{header|Haskell}}==
<
import Data.Bool (bool)
import Data.List.NonEmpty (NonEmpty, fromList, toList, unfoldr)
Line 638 ⟶ 1,217:
"\n%s is at index %d of the Calkin-Wilf sequence.\n"
(show r)
(calkinWilfIdx r)</
{{out}}
<pre>
Line 664 ⟶ 1,243:
83116 % 51639 is at index 123456789 of the Calkin-Wilf sequence.
</pre>
=={{header|J}}==
<pre>
Line 676 ⟶ 1,254:
</pre>
given definitions
<syntaxhighlight lang="j">
cw_next_term=: [: % +:@<. + -.
Line 698 ⟶ 1,276:
NB. base 2 @ reverse @ the cf's representation copies of 1 0 1 0 ...
index_cw_term=: #.@|.@(# 1 0 $~ #)@molcf@ccf
</syntaxhighlight>
Note that <code>ccf</code> could be expressed more concisely:
<syntaxhighlight lang=J>ccf=: _1 {"1 |.@(0 1 #: %@{.)^:(0~:{.)^:a:</syntaxhighlight>
=={{header|Java}}==
<syntaxhighlight lang="java">
import java.util.ArrayDeque;
import java.util.Deque;
public final class CalkinWilfSequence {
public static void main(String[] aArgs) {
Rational term = Rational.ONE;
System.out.println("First 20 terms of the Calkin-Wilf sequence are:");
for ( int i = 1; i <= 20; i++ ) {
System.out.println(String.format("%2d", i) + ": " + term);
term = nextCalkinWilf(term);
}
System.out.println();
Rational rational = new Rational(83_116, 51_639);
System.out.println(" " + rational + " is the " + termIndex(rational) + "th term of the sequence.");
}
private static Rational nextCalkinWilf(Rational aTerm) {
Rational divisor = Rational.TWO.multiply(aTerm.floor()).add(Rational.ONE).subtract(aTerm);
return Rational.ONE.divide(divisor);
}
private static long termIndex(Rational aRational) {
long result = 0;
long binaryDigit = 1;
long power = 0;
for ( long term : continuedFraction(aRational) ) {
for ( long i = 0; i < term; power++, i++ ) {
result |= ( binaryDigit << power );
}
binaryDigit = ( binaryDigit == 0 ) ? 1 : 0;
}
return result;
}
private static Deque<Long> continuedFraction(Rational aRational) {
long numerator = aRational.numerator();
long denominator = aRational.denominator();
Deque<Long> result = new ArrayDeque<Long>();
while ( numerator != 1 ) {
result.addLast(numerator / denominator);
long copyNumerator = numerator;
numerator = denominator;
denominator = copyNumerator % denominator;
}
if ( ! result.isEmpty() && result.size() % 2 == 0 ) {
final long back = result.removeLast();
result.addLast(back - 1);
result.addLast(1L);
}
return result;
}
}
final class Rational {
public Rational(long aNumerator, long aDenominator) {
if ( aDenominator == 0 ) {
throw new ArithmeticException("Denominator cannot be zero");
}
if ( aNumerator == 0 ) {
aDenominator = 1;
}
if ( aDenominator < 0 ) {
numer = -aNumerator;
denom = -aDenominator;
} else {
numer = aNumerator;
denom = aDenominator;
}
final long gcd = gcd(numer, denom);
numer = numer / gcd;
denom = denom / gcd;
}
public Rational add(Rational aOther) {
return new Rational(numer * aOther.denom + aOther.numer * denom, denom * aOther.denom);
}
public Rational subtract(Rational aOther) {
return new Rational(numer * aOther.denom - aOther.numer * denom, denom * aOther.denom);
}
public Rational multiply(Rational aOther) {
return new Rational(numer * aOther.numer, denom * aOther.denom);
}
public Rational divide(Rational aOther) {
return new Rational(numer * aOther.denom, denom * aOther.numer);
}
public Rational floor() {
return new Rational(numer / denom, 1);
}
public long numerator() {
return numer;
}
public long denominator() {
return denom;
}
@Override
public String toString() {
return numer + "/" + denom;
}
public static final Rational ONE = new Rational(1, 1);
public static final Rational TWO = new Rational(2, 1);
private long gcd(long aOne, long aTwo) {
if ( aTwo == 0 ) {
return aOne;
}
return gcd(aTwo, aOne % aTwo);
}
private long numer;
private long denom;
}
</syntaxhighlight>
{{ out }}
<pre>
First 20 terms of the Calkin-Wilf sequence are:
1: 1/1
2: 1/2
3: 2/1
4: 1/3
5: 3/2
6: 2/3
7: 3/1
8: 1/4
9: 4/3
10: 3/5
11: 5/2
12: 2/5
13: 5/3
14: 3/4
15: 4/1
16: 1/5
17: 5/4
18: 4/7
19: 7/3
20: 3/8
83116/51639 is the 123456789th term of the sequence.
</pre>
=={{header|jq}}==
'''Adapted from [[#Wren|Wren]]'''
{{works with|jq}}
'''Also works with gojq, the Go implementation of jq, and with fq'''
See [[Arithmetic/Rational#jq]] for the Rational module included by the `include` directive.
In this module, rationals are represented by JSON objects of the form {n, d}, where .n and .d are
the numerator and denominator respectively. r(n;d) is the constructor function,
and r(n;d) is pretty-printed as `n // d`.
<syntaxhighlight lang=jq>
include "rational"; # see [[Arithmetic/Rational#jq]]
### Generic Utilities
# counting from 0
def enumerate(s): foreach s as $x (-1; .+1; [., $x]);
# input string is converted from "base" to an integer, within limits
# of the underlying arithmetic operations, and without error-checking:
def to_i(base):
explode
| reverse
| map(if . > 96 then . - 87 else . - 48 end) # "a" ~ 97 => 10 ~ 87
| reduce .[] as $c
# state: [power, ans]
([1,0]; (.[0] * base) as $b | [$b, .[1] + (.[0] * $c)])
| .[1];
### The Calkin-Wilf Sequence
# Emit an array of $n terms
def calkinWilf($n):
reduce range(1;$n) as $i ( [r(1;1)];
radd(1; rminus( rmult(2; (.[$i-1]|rfloor)); .[$i-1])) as $t
| .[$i] = rdiv(r(1;1) ; $t)) ;
# input: a Rational
def toContinued:
{ a: .n,
b: .d,
res: [] }
| until( .break;
.res += [.a / .b | floor]
| (.a % .b) as $t
| .a = .b
| .b = $t
| .break = (.a == 1) )
| if .res|length % 2 == 0
then # ensure always odd
.res[-1] += -1
| .res += [1]
else .
end
| .res;
# input: an array representing a continued fraction
def getTermNumber:
reduce .[] as $n ( {b: "", d: "1"};
.b = (.d * $n) + .b
| .d = (if .d == "1" then "0" else "1" end))
| .b | to_i(2) ;
# input: a Rational in the Calkin-Wilf sequence
def getTermNumber:
reduce .[] as $n ( {b: "", d: "1"};
.b = (.d * $n) + .b
| .d = (if .d == "1" then "0" else "1" end))
| .b | to_i(2) ;
def task(r):
"The first 20 terms of the Calkin-Wilf sequence are:",
(enumerate(calkinWilf(20)[]) | "\(1+.[0]): \(.[1]|rpp)" ),
"",
"\(r|rpp) is term # \(r|toContinued|getTermNumber) of the sequence.";
task( r(83116; 51639) )
</syntaxhighlight>
'''Invocation''': jq -nrf calkin-wilf-sequence.jq
{{output}}
<pre>
The first 20 terms of the Calkin-Wilf sequence are:
1: 1 // 1
2: 1 // 2
3: 2 // 1
4: 1 // 3
5: 3 // 2
6: 2 // 3
7: 3 // 1
8: 1 // 4
9: 4 // 3
10: 3 // 5
11: 5 // 2
12: 2 // 5
13: 5 // 3
14: 3 // 4
15: 4 // 1
16: 1 // 5
17: 5 // 4
18: 4 // 7
19: 7 // 3
20: 3 // 8
83116 // 51639 is term # 123456789 of the sequence.
</pre>
=={{header|Julia}}==
{{trans|Wren}}
<
cw = zeros(Rational, n + 1)
for i in 2:n + 1
Line 738 ⟶ 1,584:
const tn = term_number(cf)
println("$r is the $tn-th term of the sequence.")
</
<pre>
The first 20 terms of the Calkin-Wilf sequence are: Rational[1//1, 1//2, 2//1, 1//3, 3//2, 2//3, 3//1, 1//4, 4//3, 3//5, 5//2, 2//5, 5//3, 3//4, 4//1, 1//5, 5//4, 4//7, 7//3, 3//8]
Line 744 ⟶ 1,590:
</pre>
=={{header|Little Man Computer}}==
Runs in a home-made simulator, which is mostly compatible with Peter Higginson's online simulator. Only, for better control of the output format, I've added an instruction OTX (extended output). To run the code in PH's simulator, replace OTX and its parameter with OUT and no parameter.
===Find first n terms===
{{trans|Pascal}}
<syntaxhighlight lang="little man computer">
// Little Man Computer, for Rosetta Code.
// Displays terms of Calkin-Wilf sequence up to the given index.
// The chosen algorithm calculates the i-th term directly from i
// (i.e. not using any previous terms).
input INP // get number of terms from user
BRZ exit // exit if 0
STA max_i // store maximum index
LDA c1 // index := 1
next_i STA i
// Write index followed by '->'
OTX 3 // non-standard: minimum width 3, no new line
LDA asc_hy
OTC
LDA asc_gt
OTC
// Find greatest power of 2 not exceeding i,
// and count the number of binary digits in i.
LDA c1
STA pwr2
loop2 STA nrDigits
LDA i
SUB pwr2
SUB pwr2
BRP double
BRA part2 // jump out if next power of 2 would exceed i
double LDA pwr2
ADD pwr2
STA pwr2
LDA nrDigits
ADD c1
BRA loop2
// The nth term a/b is calculated from the binary digits of i.
// The leading 1 is not used.
part2 LDA c1
STA a // a := 1
STA b // b := 1
LDA i
SUB pwr2
STA diff
// Pre-decrement count, since leading 1 is not used
dec_ct LDA nrDigits // count down the number of digits
SUB c1
BRZ output // if all digits done, output the result
STA nrDigits
// We now want to compare diff with pwr2/2.
// Since division is awkward in LMC, we compare 2*diff with pwr2.
LDA diff // diff := 2*diff
ADD diff
STA diff
SUB pwr2 // is diff >= pwr2 ?
BRP digit_1 // binary digit is 1 if yes, 0 if no
// If binary digit is 0 then set b := a + b
LDA a
ADD b
STA b
BRA dec_ct
// If binary digit is 1 then update diff and set a := a + b
digit_1 STA diff
LDA a
ADD b
STA a
BRA dec_ct
// Now have nth term a/b. Write it to the output.
output LDA a // write a
OTX 1 // non-standard: minimum width 1; no new line
LDA asc_sl // write slash
OTC
LDA b // write b
OTX 11 // non-standard: minimum width 1; add new line
LDA i // have we done maximum i yet?
SUB max_i
BRZ exit // if yes, exit
LDA i // if no, increment i and loop back
ADD c1
BRA next_i
exit HLT
// Constants
c1 DAT 1
asc_hy DAT 45
asc_gt DAT 62
asc_sl DAT 47
// Variables
i DAT
max_i DAT
pwr2 DAT
nrDigits DAT
diff DAT
a DAT
b DAT
// end
</syntaxhighlight>
{{out}}
<pre>
1->1/1
2->1/2
3->2/1
4->1/3
5->3/2
6->2/3
7->3/1
8->1/4
9->4/3
10->3/5
11->5/2
12->2/5
13->5/3
14->3/4
15->4/1
16->1/5
17->5/4
18->4/7
19->7/3
20->3/8
</pre>
===Find index of a given term===
{{trans|Pascal}}
The numbers in part 2 of the task are too large for LMC, so the demo program just confirms the example, that 9/4 is the 35th term.
<syntaxhighlight lang="little man computer">
// Little Man Computer, for Rosetta Code.
// Calkin-Wilf sequence: displays index of term entered by user.
INP // get numerator from user
BRZ exit // exit if 0
STA num
STA a // initialize a := numerator
INP // get denominator from user
BRZ exit // exit if 0
STA den
STA b // initialize b := denominator
LDA c0 // initialize index := 0
STA index
LDA c1 // initialize power of 2 := 1
STA pwr2
// Build binary digits of the index
loop LDA a // is a = b yet?
SUB b
BRZ break // if yes, break out of loop
BRP a_gt_b // jump if a > b
// If a < b then b := b - a, binary digit is 0
LDA b
SUB a
STA b
BRA double
// If a > b then a := a - b, binary digit is 1
a_gt_b STA a
LDA index
ADD pwr2
STA index
// In either case, on to next power of 2
double LDA pwr2
ADD pwr2
STA pwr2
BRA loop
// Out of loop, add leading binary digit 1
break LDA index
ADD pwr2
STA index
// Output the result
LDA num
OTX 1 // non-standard: minimum width = 1, no new line
LDA asc_sl
OTC
LDA den
OTX 1
LDA asc_lt // write '<-' after fraction
OTC
LDA asc_hy
OTC
LDA index
OTX 11 // non-standard: minimum width = 1, add new line
exit HLT
// Constants
c0 DAT 0
c1 DAT 1
asc_sl DAT 47
asc_lt DAT 60
asc_hy DAT 45
// Variables
num DAT
den DAT
a DAT
b DAT
pwr2 DAT
index DAT
// end
</syntaxhighlight>
{{out}}
<pre>
9/4<-35
</pre>
=={{header|Mathematica}} / {{header|Wolfram Language}}==
<
a[1] = 1;
a[n_?(GreaterThan[1])] := a[n] = 1/(2 Floor[a[n - 1]] + 1 - a[n - 1])
Line 762 ⟶ 1,802:
Break[];
]
]</
{{out}}
<pre>{1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4, 1/5, 5/4, 4/7, 7/3, 3/8}
123456789</pre>
=={{header|Maxima}}==
<syntaxhighlight lang="maxima">
/* The function fusc is related to Calkin-Wilf sequence */
fusc(n):=block(
[k:n,a:1,b:0],
while k>0 do (if evenp(k) then (k:k/2,a:a+b) else (k:(k-1)/2,b:a+b)),
b)$
/* Calkin-Wilf function using fusc */
calkin_wilf(n):=fusc(n)/fusc(n+1)$
/* Function that given a nonnegative rational returns its position in the Calkin-Wilf sequence */
cf_bin(fracti):=block(
cf_list:cf(fracti),
cf_len:length(cf_list),
if oddp(cf_len) then cf_list:reverse(cf_list) else cf_list:reverse(append(at(cf_list,[cf_list[cf_len]=cf_list[cf_len]-1]),[1])),
makelist(lambda([x],if oddp(x) then makelist(1,j,1,cf_list[x]) else makelist(0,j,1,cf_list[x]))(i),i,1,length(cf_list)), /* decoding part begins here */
apply(append,%%),
apply("+",makelist(2^i,i,0,length(%%)-1)*reverse(%%)))$
/* Test cases */
/* 20 first terms of the sequence */
makelist(calkin_wilf(i),i,1,20);
/* Position of 83116/51639 in Calkin-Wilf sequence */
83116/51639$
cf_bin(%);
</syntaxhighlight>
{{out}}
<pre>
[1,1/2,2,1/3,3/2,2/3,3,1/4,4/3,3/5,5/2,2/5,5/3,3/4,4,1/5,5/4,4/7,7/3,3/8]
123456789
</pre>
=={{header|Nim}}==
Line 771 ⟶ 1,846:
With these optimizations, the program runs in less than 1.3 s on our laptop.
<
iterator calkinWilf(): Fraction =
Line 805 ⟶ 1,880:
inc index
if an == Target: break
echo "\nThe element ", $Target, " is at position ", $index, " in the sequence."</
{{out}}
Line 813 ⟶ 1,888:
The element 83116/51639 is at position 123456789 in the sequence.</pre>
=={{header|PARI/GP}}==
{{trans|Mathematica_/_Wolfram_Language}}
<syntaxhighlight lang="PARI/GP">
\\ This function assumes the existence of a global variable 'an' for 'a[n]'
a(n) = if(n==1, 1, 1 / (2 * floor(an[n-1]) + 1 - an[n-1]));
\\ We will use a vector to hold the values and compute them iteratively to avoid stack overflow
an = vector(20);
an[1] = 1;
for(i=2, 20, an[i] = a(i));
\\ Now we print the vector
print(an);
\\ Initialize variables for the while loop
a = 1;
n = 1;
\\ Loop until the condition is met
while(a != 83116/51639,{
a = 1/(2 * floor(a) + 1 - a);
if(n>=123456789,print(n));
n++;
});
\\ Output the number of iterations needed to reach 83116/51639
print(n);
</syntaxhighlight>
{{out}}
<pre>
[1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4, 1/5, 5/4, 4/7, 7/3, 3/8]
123456789
</pre>
=={{header|Pascal}}==
These programs were written in Free Pascal, using the Lazarus IDE and the Free Pascal compiler version 3.2.0. They are based on the Wikipedia article "Calkin-Wilf tree", rather than the algorithms in the task description.
<syntaxhighlight lang="pascal">
program CWTerms;
{-------------------------------------------------------------------------------
FreePascal command-line program.
Calculates the Calkin-Wilf sequence up to the specified maximum index,
where the first term 1/1 has index 1.
Command line format is: CWTerms <max_index>
The program demonstrates 3 algorithms for calculating the sequence:
(1) Calculate term[2n] and term[2n + 1] from term[n]
(2) Calculate term[n + 1] from term[n]
(3) Calculate term[n] directly from n, without using other terms
Algorithm 1 is called first, and stores the terms in an array.
Then the program calls Algorithms 2 and 3, and checks that they agree
with Algorithm 1.
-------------------------------------------------------------------------------}
uses SysUtils;
type TRational = record
Num, Den : integer;
end;
var
terms : array of TRational;
max_index, k : integer;
// Routine to calculate array of terms up the the maiximum index
procedure CalcTerms_algo_1();
var
j, k : integer;
begin
SetLength( terms, max_index + 1);
j := 1; // index to earlier term, from which current term is calculated
k := 1; // index to current term
terms[1].Num := 1;
terms[1].Den := 1;
while (k < max_index) do begin
inc(k);
if (k and 1) = 0 then begin // or could write "if not Odd(k)"
terms[k].Num := terms[j].Num;
terms[k].Den := terms[j].Num + terms[j].Den;
end
else begin
terms[k].Num := terms[j].Num + terms[j].Den;
terms[k].Den := terms[j].Den;
inc(j);
end;
end;
end;
// Method to get each term from the preceding term.
// a/b --> b/(a + b - 2(a mod b));
function CheckTerms_algo_2() : boolean;
var
index, a, b, temp : integer;
begin
result := true;
index := 1;
a := 1;
b := 1;
while (index <= max_index) do begin
if (a <> terms[index].Num) or (b <> terms[index].Den) then
result := false;
temp := a + b - 2*(a mod b);
a := b;
b := temp;
inc( index)
end;
end;
// Mathod to calcualte each term from its index, without using other terms.
function CheckTerms_algo_3() : boolean;
var
index, a, b, pwr2, idiv2 : integer;
begin
result := true;
for index := 1 to max_index do begin
idiv2 := index div 2;
pwr2 := 1;
while (pwr2 <= idiv2) do pwr2 := pwr2 shl 1;
a := 1;
b := 1;
while (pwr2 > 1) do begin
pwr2 := pwr2 shr 1;
if (pwr2 and index) = 0 then
inc( b, a)
else
inc( a, b);
end;
if (a <> terms[index].Num) or (b <> terms[index].Den) then
result := false;
end;
end;
begin
// Read and validate maximum index
max_index := SysUtils.StrToIntDef( paramStr(1), -1); // -1 if not an integer
if (max_index <= 0) then begin
WriteLn( 'Maximum index must be a positive integer');
exit;
end;
// Calculate terms by algo 1, then check that algos 2 and 3 agree.
CalcTerms_algo_1();
if not CheckTerms_algo_2() then begin
WriteLn( 'Algorithm 2 failed');
exit;
end;
if not CheckTerms_algo_3() then begin
WriteLn( 'Algorithm 3 failed');
exit;
end;
// Display the terms
for k := 1 to max_index do
with terms[k] do
WriteLn( SysUtils.Format( '%8d: %d/%d', [k, Num, Den]));
end.
</syntaxhighlight>
{{out}}
<pre>
1: 1/1
2: 1/2
3: 2/1
4: 1/3
5: 3/2
6: 2/3
7: 3/1
8: 1/4
9: 4/3
10: 3/5
11: 5/2
12: 2/5
13: 5/3
14: 3/4
15: 4/1
16: 1/5
17: 5/4
18: 4/7
19: 7/3
20: 3/8
</pre>
<syntaxhighlight lang="pascal">
program CWIndex;
{-------------------------------------------------------------------------------
FreePascal command-line program.
Calculates index of a rational number in the Calkin-Wilf sequence,
where the first term 1/1 has index 1.
Command line format is
CWIndex <numerator> <denominator>
e.g. for the Rosetta Code example
CWIndex 83116 51639
-------------------------------------------------------------------------------}
uses SysUtils;
var
num, den : integer;
a, b : integer;
pwr2, index : qword; // 64-bit unsiged
begin
// Read and validate input.
num := SysUtils.StrToIntDef( paramStr(1), -1); // return -1 if not an integer
den := SysUtils.StrToIntDef( paramStr(2), -1);
if (num <= 0) or (den <= 0) then begin
WriteLn( 'Numerator and denominator must be positive integers');
exit;
end;
// Input OK, calculate and display index of num/den
// The index may overflow 64 bits, so turn on overflow detection
{$Q+}
a := num;
b := den;
pwr2 := 1;
index := 0;
try
while (a <> b) do begin
if (a < b) then
dec( b, a)
else begin
dec( a, b);
inc( index, pwr2);
end;
pwr2 := 2*pwr2;
end;
inc( index, pwr2);
WriteLn( SysUtils.Format( 'Index of %d/%d is %u', [num, den, index]));
except
WriteLn( 'Index is too large for 64 bits');
end;
end.
</syntaxhighlight>
{{out}}
<pre>
Index of 83116/51639 is 123456789
</pre>
=={{header|Perl}}==
{{trans|Raku}}
{{libheader|ntheory}}
<
use warnings;
use feature qw(say state);
Line 848 ⟶ 2,160:
say 'First twenty terms of the Calkin-Wilf sequence:';
printf "%s ", $calkin_wilf->next() for 1..20;
say "\n\n83116/51639 is at index: " . r2cw(83116,51639);</
{{out}}
<pre>First twenty terms of the Calkin-Wilf sequence:
Line 854 ⟶ 2,166:
83116/51639 is at index: 123456789</pre>
=={{header|Phix}}==
<!--<
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"1.0.0"</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (new even() builtin)</span>
Line 947 ⟶ 2,258:
<span style="color: #004080;">integer</span> <span style="color: #000000;">tn</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">get_term_number</span><span style="color: #0000FF;">(</span><span style="color: #000000;">cf</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%d/%d is the %,d%s term of the sequence.\n"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">&{</span><span style="color: #000000;">tn</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">ord</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tn</span><span style="color: #0000FF;">)})</span>
<!--</
{{out}}
<pre>
Line 974 ⟶ 2,285:
83116/51639 is the 123,456,789th term of the sequence.
</pre>
=={{header|Prolog}}==
<syntaxhighlight lang="prolog">
% John Devou: 26-Nov-2021
% g(N,X):- consecutively generate in X the first N elements of the Calkin-Wilf sequence
g(N,[A/B|_]-_,A/B):- N > 0.
g(N,[A/B|Ls]-[A/C,C/B|Ys],X):- N > 1, M is N-1, C is A+B, g(M,Ls-Ys,X).
g(N,X):- g(N,[1/1|Ls]-Ls,X).
% t(A/B,X):- generate in X the index of A/B in the Calkin-Wilf sequence
t(A/1,S,C,X):- X is C*(2**(A-1+S)-S).
t(A/B,S,C,X):- B > 1, divmod(A,B,M,N), T is 1-S, D is C*2**M, t(B/N,T,D,Y), X is Y + S*C*(2**M-1).
t(A/B,X):- t(A/B,1,1,X), !.
</syntaxhighlight>
{{out}}
<pre>
?- findall(X, g(20,X), L), write(L).
[1/1,1/2,2/1,1/3,3/2,2/3,3/1,1/4,4/3,3/5,5/2,2/5,5/3,3/4,4/1,1/5,5/4,4/7,7/3,3/8]
L = [1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, ... / ...|...].
?- t(83116/51639,X).
X = 123456789.
</pre>
=={{header|Python}}==
<
from math import floor
from itertools import islice, groupby
Line 1,006 ⟶ 2,341:
print('TERMS 1..20: ', ', '.join(str(x) for x in islice(cw(), 20)))
x = Fraction(83116, 51639)
print(f"\n{x} is the {get_term_num(x):_}'th term.")</
{{out}}
Line 1,012 ⟶ 2,347:
83116/51639 is the 123_456_789'th term.</pre>
=={{header|Quackery}}==
<code>cf</code> is defined at [[Continued fraction/Arithmetic/Construct from rational number#Quackery]].
<syntaxhighlight lang="quackery"> [ $ "bigrat.qky" loadfile ] now!
[ ' [ [ 1 1 ] ]
swap 1 - times
[ dup -1 peek do
2dup proper 2drop
2 * n->v
2swap -v 1 n->v v+ v+
1/v join nested join ] ] is calkin-wilf ( n --> [ )
[ 1 & ] is odd ( n --> b )
[ dup size odd not if
[ -1 split do
1 - join
1 join ] ] is oddcf ( [ --> [ )
[ 0 swap
reverse witheach
[ i odd iff
<< done
dup dip <<
bit 1 - | ] ] is rl->n ( [ --> n )
[ cf oddcf rl->n ] is cw-term ( n/d --> n )
20 calkin-wilf
witheach
[ do vulgar$ echo$ sp ]
cr cr
83116 51639 cw-term echo</syntaxhighlight>
{{out}}
<pre>1/1 1/2 2/1 1/3 3/2 2/3 3/1 1/4 4/3 3/5 5/2 2/5 5/3 3/4 4/1 1/5 5/4 4/7 7/3 3/8
123456789</pre>
=={{header|Raku}}==
In Raku, arrays are indexed from 0. The Calkin-Wilf sequence does not have a term defined at 0.
Line 1,018 ⟶ 2,393:
This implementation includes a bogus undefined value at position 0, having the bogus first term shifts the indices up by one, making the ordinal position and index match. Useful due to how reversibility function works.
<syntaxhighlight lang="raku"
# Rational to Calkin-Wilf index
Line 1,049 ⟶ 2,424:
return $num.numerator if $num.denominator == 1;
$num.nude.join: '/';
}</
{{out}}
<pre>First twenty terms of the Calkin-Wilf sequence: 1, 1/2, 2, 1/3, 3/2, 2/3, 3, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4, 1/5, 5/4, 4/7, 7/3, 3/8
Line 1,058 ⟶ 2,433:
83116/51639 is at index: 123456789</pre>
=={{header|REXX}}==
The meat of this REXX program was provided by Paul Kislanko.
<
/*────────────────────── or finds which sequence number contains a specified fraction). */
numeric digits 2000 /*be able to handle ginormic integers. */
Line 1,111 ⟶ 2,484:
obin= copies(1, f1)copies(0, f0)obin
end /*until*/
return x2d( b2x(obin) ) /*RLE2DEC: Run Length Encoding ──► decimal*/</
{{out|output|text= when using the default inputs:}}
<pre>
Line 1,118 ⟶ 2,491:
for 83116/51639, the element number for the Calkin─Wilf sequence is: 123,456,789th
</pre>
=={{header|RPL}}==
{{works with|HP|49g}}
≪ { } SWAP
'''WHILE''' DUP '''REPEAT '''
ROT OVER IDIV2
4 ROLL ROT + SWAP
'''END''' ROT DROP2
≫ '<span style="color:blue">CONTFRAC</span>' STO
≪ {1}
'''WHILE''' DUP2 SIZE > '''REPEAT'''
DUP DUP SIZE GET
DUP IP R→I 2 * 1 + SWAP - INV EVAL +
'''END''' NIP
≫ ≫ '<span style="color:blue">CWILF</span>' STO
≪
<span style="color:blue">CONTFRAC</span> DUP SIZE
'''IF''' DUP MOD '''THEN''' DROP '''ELSE'''
DUP2 GET 1 - PUT 1 + '''END'''
1 → frac pow2
≪ 0
1 frac SIZE '''FOR''' j
frac j GET
'''WHILE''' DUP '''REPEAT'''
'''IF''' j 2 MOD '''THEN''' SWAP pow2 + SWAP '''END'''
2 'pow2' STO*
1 -
'''END''' DROP
'''NEXT'''
≫ ≫ '<span style="color:blue">CWPOS</span>' STO
20 <span style="color:blue">CWILF</span>
83116 51639 <span style="color:blue">CWPOS</span>
{{out}}
<pre>
2: {1 '1/2' 2 '1/3' '3/2' '2/3' 3 '1/4' '4/3' '3/5' '5/2' '2/5' '5/3' '3/4' 4 '1/5' '5/4' '4/7' '7/3' '3/8'}
1: 123456789
</pre>
=={{header|Ruby}}==
{{trans|Python}}
<syntaxhighlight lang="ruby">cw = Enumerator.new do |y|
y << a = 1.to_r
loop { y << a = 1/(2*a.floor + 1 - a) }
end
def term_num(rat)
num, den, res, pwr, dig = rat.numerator, rat.denominator, 0, 0, 1
while den > 0
num, (digit, den) = den, num.divmod(den)
digit.times do
res |= dig << pwr
pwr += 1
end
dig ^= 1
end
res
end
puts cw.take(20).join(", ")
puts term_num (83116/51639r)
</syntaxhighlight>
<pre>1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, 1/4, 4/3, 3/5, 5/2, 2/5, 5/3, 3/4, 4/1, 1/5, 5/4, 4/7, 7/3, 3/8
123456789
</pre>
=={{header|Rust}}==
<
// num = "0.3"
Line 1,174 ⟶ 2,613:
let r = Rational::new(83116, 51639);
println!("{} is the {}th term of the sequence.", r, term_number(&r));
}</
{{out}}
Line 1,201 ⟶ 2,640:
83116/51639 is the 123456789th term of the sequence.
</pre>
=={{header|Scheme}}==
{{works with|Chez Scheme}}
'''Continued Fraction support'''
<syntaxhighlight lang="scheme">; Create a terminating Continued Fraction generator for the given rational number.
; Returns one term per call; returns #f when no more terms remaining.
(define make-continued-fraction-gen
(lambda (rat)
(let ((num (numerator rat)) (den (denominator rat)))
(lambda ()
(if (= den 0)
#f
(let ((ret (quotient num den))
(rem (modulo num den)))
(set! num den)
(set! den rem)
ret))))))
; Return the continued fraction representation of a rational number as a list of terms.
(define rat->cf-list
(lambda (rat)
(let ((cf (make-continued-fraction-gen rat))
(lst '()))
(let loop ((term (cf)))
(when term
(set! lst (append lst (list term)))
(loop (cf))))
lst)))
; Enforce the length of the given continued fraction list to be odd.
; Changes the list in situ (if needed), and returns its possibly changed value.
(define continued-fraction-list-enforce-odd-length!
(lambda (cf)
(when (even? (length cf))
(let ((cf-last-cons (list-tail cf (1- (length cf)))))
(set-car! cf-last-cons (1- (car cf-last-cons)))
(set-cdr! cf-last-cons (cons 1 '()))))
cf))</syntaxhighlight>
'''Calkin-Wilf sequence'''
<syntaxhighlight lang="scheme">; Create a Calkin-Wilf sequence generator.
(define make-calkin-wilf-gen
(lambda ()
(let ((an 1))
(lambda ()
(let ((ret an))
(set! an (/ 1 (+ (* 2 (floor an)) 1 (- an))))
ret)))))
; Return the position in the Calkin-Wilf sequence of the given rational number.
(define calkin-wilf-position
(lambda (rat)
; Run-length encodes binary value. Assumes first run is 1's. Args: initial value,
; starting place value (a power of 2), and list of run lengths (list must be odd length).
(define encode-list-of-runs
(lambda (value placeval lstruns)
; Encode a single run in a binary value. Args: initial value, bit value (0 or 1),
; starting place value (a power of 2), number of places (bits) to encode.
; Returns multiple values: the encoded value, and the new place value.
(define encode-run
(lambda (value bitval placeval places)
(if (= places 1)
(values (+ value (* bitval placeval)) (* 2 placeval))
(encode-run (+ value (* bitval placeval)) bitval (* 2 placeval) (1- places)))))
; Loop through the list of runs two at a time. If list of length 1, do a final
; '1'-bit encode and return the value. Otherwise, do a '1'-bit then '0'-bit encode,
; and recurse to do the next two runs.
(let-values (((value-1 placeval-1) (encode-run value 1 placeval (car lstruns))))
(if (= 1 (length lstruns))
value-1
(let-values (((value-2 placeval-2) (encode-run value-1 0 placeval-1 (cadr lstruns))))
(encode-list-of-runs value-2 placeval-2 (cddr lstruns)))))))
; Return the run-length binary encoding from the odd-length Calkin-Wilf sequence of the
; given rational number. This is equal to the number's position in the sequence.
(encode-list-of-runs 0 1 (continued-fraction-list-enforce-odd-length! (rat->cf-list rat)))))</syntaxhighlight>
'''The Task'''
<syntaxhighlight lang="scheme">(let ((count 20)
(cw (make-calkin-wilf-gen)))
(printf "~%First ~a terms of the Calkin-Wilf sequence:~%" count)
(do ((num 1 (1+ num)))
((> num count))
(printf "~2d : ~a~%" num (cw))))
(printf "~%Positions in Calkin-Wilf sequence of given numbers:~%")
(let ((num 9/4))
(printf "~a @ ~a~%" num (calkin-wilf-position num)))
(let ((num 83116/51639))
(printf "~a @ ~a~%" num (calkin-wilf-position num)))</syntaxhighlight>
{{out}}
<pre>
First 20 terms of the Calkin-Wilf sequence:
1 : 1
2 : 1/2
3 : 2
4 : 1/3
5 : 3/2
6 : 2/3
7 : 3
8 : 1/4
9 : 4/3
10 : 3/5
11 : 5/2
12 : 2/5
13 : 5/3
14 : 3/4
15 : 4
16 : 1/5
17 : 5/4
18 : 4/7
19 : 7/3
20 : 3/8
Positions in Calkin-Wilf sequence of given numbers:
9/4 @ 35
83116/51639 @ 123456789
</pre>
=={{header|Sidef}}==
<
return 1 if (n == 1)
1/(2*floor(__FUNC__(n-1)) + 1 - __FUNC__(n-1))
Line 1,223 ⟶ 2,775:
with (83116/51639) {|r|
say ("\n#{r.as_rat} is at index: ", r2cw(r))
}</
{{out}}
<pre>
Line 1,230 ⟶ 2,782:
83116/51639 is at index: 123456789
</pre>
=={{header|V (Vlang)}}==
{{trans|Go}}s.
<syntaxhighlight lang="v (vlang)">import math.fractions
import math
import strconv
fn calkin_wilf(n int) []fractions.Fraction {
mut cw := []fractions.Fraction{len: n+1}
cw[0] = fractions.fraction(1, 1)
one := fractions.fraction(1, 1)
two := fractions.fraction(2, 1)
for i in 1..n {
mut t := cw[i-1]
mut f := t.f64()
f = math.floor(f)
t = fractions.approximate(f)
t*=two
t-= cw[i-1]
t+=one
t=t.reciprocal()
cw[i] = t
}
return cw
}
fn to_continued(r fractions.Fraction) []int {
idx := r.str().index('/') or {0}
mut a := r.str()[..idx].i64()
mut b := r.str()[idx+1..].i64()
mut res := []int{}
for {
res << int(a/b)
t := a % b
a, b = b, t
if a == 1 {
break
}
}
le := res.len
if le%2 == 0 { // ensure always odd
res[le-1]--
res << 1
}
return res
}
fn get_term_number(cf []int) ?int {
mut b := ""
mut d := "1"
for n in cf {
b = d.repeat(n)+b
if d == "1" {
d = "0"
} else {
d = "1"
}
}
i := strconv.parse_int(b, 2, 64)?
return int(i)
}
fn commatize(n int) string {
mut s := "$n"
if n < 0 {
s = s[1..]
}
le := s.len
for i := le - 3; i >= 1; i -= 3 {
s = s[0..i] + "," + s[i..]
}
if n >= 0 {
return s
}
return "-" + s
}
fn main() {
cw := calkin_wilf(20)
println("The first 20 terms of the Calkin-Wilf sequnence are:")
for i := 1; i <= 20; i++ {
println("${i:2}: ${cw[i-1]}")
}
println('')
r := fractions.fraction(83116, 51639)
cf := to_continued(r)
tn := get_term_number(cf) or {0}
println("$r is the ${commatize(tn)}th term of the sequence.")
}</syntaxhighlight>
{{out}}
<pre>
The first 20 terms of the Calkin-Wilf sequnence are:
1: 1/1
2: 1/2
3: 2/1
4: 1/3
5: 3/2
6: 2/3
7: 3/1
8: 1/4
9: 4/3
10: 3/5
11: 5/2
12: 2/5
13: 5/3
14: 3/4
15: 4/1
16: 1/5
17: 5/4
18: 4/7
19: 7/3
20: 3/8
83116/51639 is the 123,456,789th term of the sequence.
</pre>
Line 1,235 ⟶ 2,902:
{{libheader|Wren-rat}}
{{libheader|Wren-fmt}}
<
import "./fmt" for Fmt, Conv
var calkinWilf = Fn.new { |n|
Line 1,284 ⟶ 2,951:
var cf = toContinued.call(r)
var tn = getTermNumber.call(cf)
Fmt.print("$s is the $,r term of the sequence.", r, tn)</
{{out}}
| |||