View source for Calkin-Wilf sequence

{{task}}

The '''Calkin-Wilf sequence''' contains every nonnegative rational number exactly once. 

It can be calculated recursively as follows:

       <big> {{math|a<sub>1</sub>}}   =  {{math|1}} </big>
       <big> {{math|a<sub>n+1</sub>}}  =  {{math|1/(2&lfloor;a<sub>n</sub>&rfloor;+1-a<sub>n</sub>)}} for n > 1 </big>


;Task part 1:
* Show on this page terms 1 through 20 of the Calkin-Wilf sequence.

<br>To avoid floating point error, you may want to use a rational number data type.


It is also possible, given a non-negative rational number, to determine where it appears in the sequence without calculating the sequence. The procedure is to get the continued fraction representation of the rational and use it as the run-length encoding of the binary representation of the term number, beginning from the end of the continued fraction. 
It only works if the number of terms in the continued fraction is odd- use either of the two equivalent representations to achieve this:

       <big> {{math|[a<sub>0</sub>; a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>n</sub>]}}  =  {{math|[a<sub>0</sub>; a<sub>1</sub>, a<sub>2</sub> ,..., a<sub>n</sub>-1, 1]}} </big>


;Example:
The fraction &nbsp; '''9/4''' &nbsp; has odd continued fraction representation &nbsp; &nbsp; <big> {{math|2; 3, 1}},</big> &nbsp; &nbsp; giving a binary representation of &nbsp; '''100011''', 
<br>which means &nbsp; '''9/4''' &nbsp; appears as the &nbsp; '''35th''' &nbsp; term of the sequence.


;Task part 2:
* Find the position of the number &nbsp; <big>'''<sup>83116</sup>'''<big>'''/'''</big>'''<sub>51639</sub>'''</big> &nbsp; in the Calkin-Wilf sequence.

;Related tasks:
:* &nbsp; [[Fusc sequence]].

;See also:
* Wikipedia entry:  [[wp:Calkin%E2%80%93Wilf_tree|Calkin-Wilf tree]]
* [[Continued fraction]]
* [[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}}==
<syntaxhighlight lang="applescript">-- Return the first n terms of the sequence. Tree generation. Faster for this purpose.
on CalkinWilfSequence(n)
    script o
        property sequence : {{1, 1}} -- Initialised with the first term ({numerator, denominator}).
    end script
    
    -- Work through the growing sequence list, adding the two children of each term to the end and
    -- converting each term to text representing the vulgar fraction. Stop adding children halfway through.
    set halfway to n div 2
    repeat with position from 1 to n
        set {numerator, denominator} to item position of o's sequence
        if (position ≤ halfway) then
            tell numerator + denominator
                set end of o's sequence to {numerator, it}
                if ((position < halfway) or (position * 2 < n)) then set end of o's sequence to {it, denominator}
            end tell
        end if
        set item position of o's sequence to (numerator as text) & "/" & denominator
    end repeat
    
    return o's sequence
end CalkinWilfSequence

-- Alternatively, return terms pos1 to pos2. Binary run-length encoding. Doesn't need to work from the beginning of the sequence.
on CalkinWilfSequence2(pos1, pos2)
    script o
        property sequence : {}
    end script
    
    repeat with position from pos1 to pos2
        -- Build a continued fraction list from the binary run-length encoding of this position index.
        -- There's no need to put the last value into the list as it's used immediately.
        set continuedFraction to {}
        set bitValue to 1
        set runLength to 0
        repeat until (position = 0)
            if (position mod 2 = bitValue) then
                set runLength to runLength + 1
            else
                set end of continuedFraction to runLength
                set bitValue to (bitValue + 1) mod 2
                set runLength to 1
            end if
            set position to position div 2
        end repeat
        -- Work out the numerator and denominator from the continued fraction and derive text representing the vulgar fraction.
        set numerator to runLength
        set denominator to 1
        repeat with i from (count continuedFraction) to 1 by -1
            tell numerator
                set numerator to numerator * (item i of continuedFraction) + denominator
                set denominator to it
            end tell
        end repeat
        set end of o's sequence to (numerator as text) & "/" & denominator
    end repeat
    
    return o's sequence
end CalkinWilfSequence2

-- Return the sequence position of the term with the given numerator and denominator.
on CalkinWilfSequencePosition(numerator, denominator)
    -- Build a continued fraction list from the input.
    set continuedFraction to {}
    repeat until (denominator is 0)
        set end of continuedFraction to numerator div denominator
        set {numerator, denominator} to {denominator, numerator mod denominator}
    end repeat
    -- If it has an even number of entries, convert to the equivalent odd number.
    if ((count continuedFraction) mod 2 is 0) then
        set last item of continuedFraction to (last item of continuedFraction) - 1
        set end of continuedFraction to 1
    end if
    -- "Binary run-length decode" the entries to get the position index.
    set position to 0
    set bitValue to 1
    repeat with i from (count continuedFraction) to 1 by -1
        repeat (item i of continuedFraction) times
            set position to position * 2 + bitValue
        end repeat
        set bitValue to (bitValue + 1) mod 2
    end repeat
    
    return position
end CalkinWilfSequencePosition

-- Task code:
local sequenceResult1, sequenceResult2, positionResult, output, astid
set sequenceResult1 to CalkinWilfSequence(20)
set sequenceResult2 to CalkinWilfSequence2(1, 20)
set positionResult to CalkinWilfSequencePosition(83116, 51639)
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to ", "
set output to "First twenty terms of sequence using tree generation:" & (linefeed & sequenceResult1)
set output to output & (linefeed & "Ditto using binary run-length encoding:") & (linefeed & sequenceResult1)
set AppleScript's text item delimiters to astid
set output to output & (linefeed & "83116/51639 is term number " & positionResult)
return output</syntaxhighlight>

{{output}}
<syntaxhighlight lang="applescript">"First twenty terms of sequence using tree generation:
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"</syntaxhighlight>
=={{header|Arturo}}==

<syntaxhighlight lang="rebol">n: new 1
d: new 1
calkinWilf: function [] .export:[n,d] [
    n: (d - n) + 2 * (n/d) * d 
    tmp: d
    d: n
    n: tmp
    return @[n d]
]

first20: [[1 1]] ++ map 1..19 => calkinWilf
print "The first 20 terms of the Calkwin-Wilf sequence are:"
print map first20 'f -> ~"|f\0|/|f\1|"

n: new 1
d: new 1
indx: new 1

target: [83116, 51639]

while ø [
    inc 'indx
    if target = calkinWilf -> break
]

print ""
print ["The element" ~"|target\0|/|target\1|" "is at position" indx "in the sequence."]</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 

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=R0NEIOKGkCB7bSDwnZWK4o2fKDA8beKGkPCdlah88J2VqSkg8J2VqH0KX3doaWxlXyDihpAge/CdlL3ijZ/wnZS+4oiY8J2UvV/wnZWjX/CdlL7iiJjwnZS94o2f8J2UvvCdlal9ClNpbSDihpAgewogIHjwnZWKMTog8J2VqOKAvzE7CiAgMPCdlYp5OiAw4oC/8J2VqTsKICDijIrwnZWo4oC/8J2VqSDDtyDwnZWoIEdDRCDwnZWpCn0KQWRkIOKGkCB7CiAgMOKAv2Ig8J2ViiDwnZWpOiDwnZWpOwogIPCdlagg8J2ViiAw4oC/eTog8J2VqDsKICBh4oC/YiDwnZWKIHjigL95OgogICgoYcOXeSkreMOXYikgU2ltIGLDl3kKfQpOZXh0IOKGkCB7buKAv2Q6IOKMvSgyw5fijIrDt8K0buKAv2Qp4oC/MSBBZGQgKGQtbinigL9kfQpDYWwg4oaQIHtOZXh04o2f8J2VqSAx4oC/MX0KCuKAolNob3cgQ2FsIDEr4oaVMjA= 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}}
<syntaxhighlight lang="cpp">#include <iostream>
#include <vector>
#include <boost/rational.hpp>

using rational = boost::rational<unsigned long>;

unsigned long floor(const rational& r) {
    return r.numerator()/r.denominator();
}

rational calkin_wilf_next(const rational& term) {
    return 1UL/(2UL * floor(term) + 1UL - term);
}

std::vector<unsigned long> continued_fraction(const rational& r) {
    unsigned long a = r.numerator();
    unsigned long b = r.denominator();
    std::vector<unsigned long> result;
    do {
        result.push_back(a/b);
        unsigned long c = a;
        a = b;
        b = c % b;
    } while (a != 1);
    if (result.size() > 0 && result.size() % 2 == 0) {
        --result.back();
        result.push_back(1);
    }
    return result;
}

unsigned long term_number(const rational& r) {
    unsigned long result = 0;
    unsigned long d = 1;
    unsigned long p = 0;
    for (unsigned long n : continued_fraction(r)) {
        for (unsigned long i = 0; i < n; ++i, ++p)
            result |= (d << p);
        d = !d;
    }
    return result;
}

int main() {
    rational term = 1;
    std::cout << "First 20 terms of the Calkin-Wilf sequence are:\n";
    for (int i = 1; i <= 20; ++i) {
        std::cout << std::setw(2) << i << ": " << term << '\n';
        term = calkin_wilf_next(term);
    }
    rational r(83116, 51639);
    std::cout << r << " is the " << term_number(r) << "th term of the sequence.\n";
}</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|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===
<syntaxhighlight lang="fsharp">
// 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
<syntaxhighlight lang="fsharp">
cW |> Seq.take 20 |> Seq.iter(fun(n,g)->printf "%d/%d " n g);printfn ""
</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
</pre>
; Indexof 83116/51639
<syntaxhighlight lang="fsharp">
printfn "%d" (1+Seq.findIndex(fun n->n=(83116,51639)) cW)
</syntaxhighlight>
{{out}}
<pre>
123456789
</pre>
=={{header|Factor}}==
{{works with|Factor|0.99 2020-08-14}}
<syntaxhighlight lang="factor">USING: formatting io kernel lists lists.lazy math
math.continued-fractions math.functions math.parser prettyprint
sequences strings vectors ;

: next-cw ( x -- y ) [ floor dup + ] [ 1 swap - + recip ] bi ;

: calkin-wilf ( -- list ) 1 [ next-cw ] lfrom-by ;

: >continued-fraction ( x -- seq )
    1vector [ dup last integer? ] [ dup next-approx ] until
    dup length even? [ unclip-last 1 - suffix! 1 suffix! ] when ;

: cw-index ( x -- n )
    >continued-fraction <reversed>
    [ even? CHAR: 1 CHAR: 0 ? <string> ] map-index concat bin> ;

! Task
"First 20 terms of the Calkin-Wilf sequence:" print
20 calkin-wilf ltake [ pprint bl ] leach nl nl

83116/51639 cw-index "83116/51639 is at index %d.\n" printf</syntaxhighlight>
{{out}}
<pre>
First 20 terms of the Calkin-Wilf sequence:
1 1/2 2 1/3 1+1/2 2/3 3 1/4 1+1/3 3/5 2+1/2 2/5 1+2/3 3/4 4 1/5 1+1/4 4/7 2+1/3 3/8 

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.

<syntaxhighlight lang="freebasic">#include "gcd.bas"

type rational
    num as integer
    den as integer
end type

dim shared as rational ONE, TWO
ONE.num = 1 : ONE.den = 1
TWO.num = 2 : TWO.den = 1

function simplify( byval a as rational ) as rational
   dim as uinteger g = gcd( a.num, a.den )
   a.num /= g : a.den /= g
   if a.den < 0 then
       a.den = -a.den
       a.num = -a.num
   end if
   return a
end function

operator + ( a as rational, b as rational ) as rational
    dim as rational ret
    ret.num = a.num * b.den + b.num*a.den
    ret.den = a.den * b.den
    return simplify(ret)
end operator

operator - ( a as rational, b as rational ) as rational
    dim as rational ret
    ret.num = a.num * b.den - b.num*a.den
    ret.den = a.den * b.den
    return simplify(ret)
end operator

operator * ( a as rational, b as rational ) as rational
    dim as rational ret
    ret.num = a.num * b.num
    ret.den = a.den * b.den
    return simplify(ret)
end operator

operator / ( a as rational, b as rational ) as rational
    dim as rational ret
    ret.num = a.num * b.den
    ret.den = a.den * b.num
    return simplify(ret)
end operator

function floor( a as rational ) as rational
    dim as rational ret
    ret.den = 1
    ret.num = a.num \ a.den
    return ret
end function

function cw_nextterm( q as rational ) as rational
    dim as rational ret = (TWO*floor(q))
    ret = ret + ONE : ret = ret - q 
    return ONE / ret
end function

function frac_to_int( byval a as rational ) as uinteger
    redim as uinteger cfrac(-1)
    dim as integer  lt = -1, ones = 1, ret = 0
    do
        lt += 1
        redim preserve as uinteger cfrac(0 to lt)
        cfrac(lt) = floor(a).num
        a = a - floor(a) : a = ONE / a
    loop until a.num = 0 or a.den = 0
    if lt mod 2 = 1 and cfrac(lt) = 1 then
        lt -= 1
        cfrac(lt)+=1
        redim preserve as uinteger cfrac(0 to lt)
    end if
    if lt mod 2 = 1 and cfrac(lt) > 1 then
        cfrac(lt) -= 1
        lt += 1
        redim preserve as uinteger cfrac(0 to lt)
        cfrac(lt) = 1
    end if
    for i as integer = lt to 0 step -1
        for j as integer = 1 to cfrac(i)
            ret *= 2
            if ones = 1 then  ret += 1
        next j
        ones = 1 - ones
    next i
    return ret
end function

function disp_rational( a as rational ) as string
    if a.den = 1 or a.num= 0 then return str(a.num)
    return str(a.num)+"/"+str(a.den)
end function

dim as rational q
q.num = 1
q.den = 1
for i as integer = 1 to 20
    print i, disp_rational(q)
    q = cw_nextterm(q)
next i

q.num = 83116
q.den = 51639
print disp_rational(q)+" is the "+str(frac_to_int(q))+"th term."</syntaxhighlight>

{{out}}
<pre>
 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
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.
<syntaxhighlight lang="go">package main

import (
    "fmt"
    "math"
    "math/big"
    "strconv"
    "strings"
)

func calkinWilf(n int) []*big.Rat {
    cw := make([]*big.Rat, n+1)
    cw[0] = big.NewRat(1, 1)
    one := big.NewRat(1, 1)
    two := big.NewRat(2, 1)
    for i := 1; i < n; i++ {
        t := new(big.Rat).Set(cw[i-1])
        f, _ := t.Float64()
        f = math.Floor(f)
        t.SetFloat64(f)
        t.Mul(t, two)
        t.Sub(t, cw[i-1])
        t.Add(t, one)
        t.Inv(t)
        cw[i] = new(big.Rat).Set(t)
    }
    return cw
}

func toContinued(r *big.Rat) []int {
    a := r.Num().Int64()
    b := r.Denom().Int64()
    var res []int
    for {
        res = append(res, int(a/b))
        t := a % b
        a, b = b, t
        if a == 1 {
            break
        }
    }
    le := len(res)
    if le%2 == 0 { // ensure always odd
        res[le-1]--
        res = append(res, 1)
    }
    return res
}

func getTermNumber(cf []int) int {
    b := ""
    d := "1"
    for _, n := range cf {
        b = strings.Repeat(d, n) + b
        if d == "1" {
            d = "0"
        } else {
            d = "1"
        }
    }
    i, _ := strconv.ParseInt(b, 2, 64)
    return int(i)
}

func commatize(n int) string {
    s := fmt.Sprintf("%d", n)
    if n < 0 {
        s = s[1:]
    }
    le := len(s)
    for i := le - 3; i >= 1; i -= 3 {
        s = s[0:i] + "," + s[i:]
    }
    if n >= 0 {
        return s
    }
    return "-" + s
}

func main() {
    cw := calkinWilf(20)
    fmt.Println("The first 20 terms of the Calkin-Wilf sequnence are:")
    for i := 1; i <= 20; i++ {
        fmt.Printf("%2d: %s\n", i, cw[i-1].RatString())
    }
    fmt.Println()
    r := big.NewRat(83116, 51639)
    cf := toContinued(r)
    tn := getTermNumber(cf)
    fmt.Printf("%s is the %sth term of the sequence.\n", r.RatString(), commatize(tn))
}</syntaxhighlight>

{{out}}
<pre>
The first 20 terms of the Calkin-Wilf sequnence are:
 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

83116/51639 is the 123,456,789th term of the sequence.
</pre>
=={{header|Haskell}}==
<syntaxhighlight lang="haskell">import Control.Monad (forM_)
import Data.Bool (bool)
import Data.List.NonEmpty (NonEmpty, fromList, toList, unfoldr)
import Text.Printf (printf)

-- The infinite Calkin-Wilf sequence, a(n), starting with a(1) = 1.
calkinWilfs :: [Rational]
calkinWilfs = iterate (recip . succ . ((-) =<< (2 *) . fromIntegral . floor)) 1

-- The index into the Calkin-Wilf sequence of a given rational number, starting
-- with 1 at index 1.
calkinWilfIdx :: Rational -> Integer
calkinWilfIdx = rld . cfo

-- A continued fraction representation of a given rational number, guaranteed
-- to have an odd length.
cfo :: Rational -> NonEmpty Int
cfo = oddLen . cf

-- The canonical (i.e. shortest) continued fraction representation of a given
-- rational number.
cf :: Rational -> NonEmpty Int
cf = unfoldr step
  where
    step r =
      case properFraction r of
        (n, 1) -> (succ n, Nothing)
        (n, 0) -> (n, Nothing)
        (n, f) -> (n, Just (recip f))

-- Ensure a continued fraction has an odd length.
oddLen :: NonEmpty Int -> NonEmpty Int
oddLen = fromList . go . toList
  where
    go [x, y] = [x, pred y, 1]
    go (x:y:zs) = x : y : go zs
    go xs = xs

-- Run-length decode a continued fraction.
rld :: NonEmpty Int -> Integer
rld = snd . foldr step (True, 0)
  where
    step i (b, n) =
      let p = 2 ^ i
      in (not b, n * p + bool 0 (pred p) b)

main :: IO ()
main = do
  forM_ (take 20 $ zip [1 :: Int ..] calkinWilfs) $
    \(i, r) -> printf "%2d  %s\n" i (show r)
  let r = 83116 / 51639
  printf
    "\n%s is at index %d of the Calkin-Wilf sequence.\n"
    (show r)
    (calkinWilfIdx r)</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

83116 % 51639 is at index 123456789 of the Calkin-Wilf sequence.
</pre>
=={{header|J}}==
<pre>
   cw_next_term^:(<20)1x
1 1r2 2 1r3 3r2 2r3 3 1r4 4r3 3r5 5r2 2r5 5r3 3r4 4 1r5 5r4 4r7 7r3 3r8

   (,. index_cw_term&>) 3r4 53r37 83116r51639
        3r4        14
      53r37      1081
83116r51639 123456789
</pre>
given definitions
<syntaxhighlight lang="j">
cw_next_term=: [: % +:@<. + -.

ccf =: compute_continued_fraction=: 3 :0
 if. 0 -: y do.
  , 0
 else.
  result=. i. 0
  remainder=. % y
  whilst. remainder do.
   remainder=. % remainder
   integer_part=. <. remainder
   remainder=. remainder - integer_part
   result=. result , integer_part
  end.
 end.
)

molcf =: make_odd_length_continued_fraction=: (}: , 1 ,~ <:@{:)^:(0 -: 2 | #)

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}}
<syntaxhighlight lang="julia">function calkin_wilf(n)
    cw = zeros(Rational, n + 1)
    for i in 2:n + 1
        t = Int(floor(cw[i - 1])) * 2 - cw[i - 1] + 1
        cw[i] = 1 // t
    end
    return cw[2:end]
end

function continued(r::Rational)
    a, b = r.num, r.den
    res = []
    while true
        push!(res, Int(floor(a / b)))
        a, b = b, a % b
        a == 1 && break
    end
    return res
end

function term_number(cf)
    b, d = "", "1"
    for n in cf
        b = d^n * b
        d = (d == "1") ? "0" : "1"
    end
    return parse(Int, b, base=2)
end

const cw = calkin_wilf(20)
println("The first 20 terms of the Calkin-Wilf sequence are: $cw")

const r = 83116 // 51639
const cf = continued(r)
const tn = term_number(cf)
println("$r is the $tn-th term of the sequence.")
</syntaxhighlight>{{out}}
<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]
83116//51639 is the 123456789-th term of the sequence.
</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}}==
<syntaxhighlight lang="mathematica">ClearAll[a]
a[1] = 1;
a[n_?(GreaterThan[1])] := a[n] = 1/(2 Floor[a[n - 1]] + 1 - a[n - 1])
a /@ Range[20]

ClearAll[a]
a = 1;
n = 1;
Dynamic[n]
done = False;
While[! done,
 a = 1/(2 Floor[a] + 1 - a);
 n++;
 If[a == 83116/51639,
  Print[n];
  Break[];
  ]
 ]</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|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}}==
We ignored the standard module “rationals” which is slow and have rather defined a fraction as a tuple of two 32 bits unsigned integers (slightly faster than 64 bits signed integers and sufficient for this task). Moreover, we didn’t do operations on fractions and computed directly the numerator and denominator values at each step. The fractions built this way are irreducible (which avoids to compute a GCD which is a slow operation).
With these optimizations, the program runs in less than 1.3 s on our laptop.

<syntaxhighlight lang="nim">type Fraction = tuple[num, den: uint32]

iterator calkinWilf(): Fraction =
  ## Yield the successive values of the sequence.
  var n, d = 1u32
  yield (n, d)
  while true:
    n = 2 * (n div d) * d + d - n
    swap n, d
    yield (n, d)

proc `$`(fract: Fraction): string =
  ## Return the representation of a fraction.
  $fract.num & '/' & $fract.den

func `==`(a, b: Fraction): bool {.inline.} =
  ## Compare two fractions. Slightly faster than comparison of tuples.
  a.num == b.num and a.den == b.den

when isMainModule:

  echo "The first 20 terms of the Calkwin-Wilf sequence are:"
  var count = 0
  for an in calkinWilf():
    inc count
    stdout.write $an & ' '
    if count == 20: break
  stdout.write '\n'

  const Target: Fraction = (83116u32, 51639u32)
  var index = 0
  for an in calkinWilf():
    inc index
    if an == Target: break
  echo "\nThe element ", $Target, " is at position ", $index, " in the sequence."</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 

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}}
<syntaxhighlight lang="perl">use strict;
use warnings;
use feature qw(say state);

use ntheory      'fromdigits';
use List::Lazy   'lazy_list';
use Math::AnyNum ':overload';

my $calkin_wilf = lazy_list { state @cw = 1; push @cw, 1 / ( (2 * int $cw[0]) + 1 - $cw[0] ); shift @cw };

sub r2cf {
  my($num, $den) = @_;
  my($n, @cf);
  my $f = sub { return unless $den;
               my $q = int($num/$den);
               ($num, $den) = ($den, $num - $q*$den);
               $q;
             };
  push @cf, $n while defined($n = $f->());
  reverse @cf;
}

sub r2cw {
    my($num, $den) = @_;
    my $bits;
    my @f = r2cf($num, $den);
    $bits .= ($_%2 ? 0 : 1) x $f[$_] for 0..$#f;
    fromdigits($bits, 2);
}

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);</syntaxhighlight>
{{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

83116/51639 is at index: 123456789</pre>
=={{header|Phix}}==
<!--<syntaxhighlight lang="phix">(phixonline)-->
 <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>
 
 <span style="color: #008080;">function</span> <span style="color: #000000;">calkin_wilf</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">len</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">sequence</span> <span style="color: #000000;">cw</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">len</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">len</span> <span style="color: #008080;">do</span>
         <span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,(</span><span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">/</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">2</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">d</span><span style="color: #0000FF;">-</span><span style="color: #000000;">n</span><span style="color: #0000FF;">}</span>
         <span style="color: #000000;">cw</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">}</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">cw</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
 
 <span style="color: #008080;">function</span> <span style="color: #000000;">odd_length</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">cf</span><span style="color: #0000FF;">)</span>
     <span style="color: #000080;font-style:italic;">-- replace even length continued fraction with odd length equivalent
 --  if remainder(length(cf),2)=0 then</span>
     <span style="color: #008080;">if</span> <span style="color: #7060A8;">even</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">cf</span><span style="color: #0000FF;">))</span> <span style="color: #008080;">then</span>
         <span style="color: #000000;">cf</span><span style="color: #0000FF;">[$]</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span>
         <span style="color: #000000;">cf</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">1</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">cf</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">to_continued_fraction</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">integer</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">r</span>
     <span style="color: #004080;">sequence</span> <span style="color: #000000;">cf</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>
     <span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
         <span style="color: #000000;">cf</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">/</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
         <span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #7060A8;">remainder</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)}</span>
         <span style="color: #008080;">if</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
     <span style="color: #000000;">cf</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">odd_length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">cf</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">cf</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">get_term_number</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">cf</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">sequence</span> <span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">cf</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
         <span style="color: #000000;">b</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span><span style="color: #000000;">cf</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])</span>
         <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">-</span><span style="color: #000000;">d</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">bits_to_int</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">t</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #000080;font-style:italic;">-- additional verification methods (2 of)</span>
 <span style="color: #008080;">function</span> <span style="color: #000000;">i_to_cf</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
 <span style="color: #000080;font-style:italic;">--  sequence b = trim_tail(int_to_bits(i,32),0)&2,</span>
     <span style="color: #004080;">sequence</span> <span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">int_to_bits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)&</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span>
              <span style="color: #000000;">cf</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #000000;">0</span><span style="color: #0000FF;">?{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">}:{})</span>
     <span style="color: #008080;">while</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)></span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
         <span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
             <span style="color: #008080;">if</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]!=</span><span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">then</span>
                 <span style="color: #000000;">cf</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span>
                 <span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">..$]</span>
                 <span style="color: #008080;">exit</span>
             <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
     <span style="color: #000000;">cf</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">odd_length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">cf</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">cf</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">cf2r</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">cf</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">cf</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">to</span> <span style="color: #000000;">2</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
         <span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">+</span><span style="color: #000000;">d</span><span style="color: #0000FF;">*</span><span style="color: #000000;">cf</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]}</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">+</span><span style="color: #000000;">cf</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">}</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">prettyr</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">integer</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">r</span>
     <span style="color: #008080;">return</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span><span style="color: #0000FF;">?</span><span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%d"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">):</span><span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%d/%d"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">}))</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #004080;">sequence</span> <span style="color: #000000;">cw</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">calkin_wilf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">20</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;">"The first 20 terms of the Calkin-Wilf sequence are:\n"</span><span style="color: #0000FF;">)</span>
 <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">20</span> <span style="color: #008080;">do</span>
     <span style="color: #004080;">string</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">prettyr</span><span style="color: #0000FF;">(</span><span style="color: #000000;">cw</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]),</span>
            <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">prettyr</span><span style="color: #0000FF;">(</span><span style="color: #000000;">cf2r</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i_to_cf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)))</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">get_term_number</span><span style="color: #0000FF;">(</span><span style="color: #000000;">to_continued_fraction</span><span style="color: #0000FF;">(</span><span style="color: #000000;">cw</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</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;">"%2d: %-4s [==&gt; %2d: %-3s]\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">})</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">for</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;">"\n"</span><span style="color: #0000FF;">)</span>
 <span style="color: #004080;">sequence</span> <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">83116</span><span style="color: #0000FF;">,</span><span style="color: #000000;">51639</span><span style="color: #0000FF;">}</span>
 <span style="color: #004080;">sequence</span> <span style="color: #000000;">cf</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">to_continued_fraction</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)</span>
 <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>
<!--</syntaxhighlight>-->
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<pre>
The first 20 terms of the Calkin-Wilf sequence are:
 1: 1    [==>  1: 1  ]
 2: 1/2  [==>  2: 1/2]
 3: 2    [==>  3: 2  ]
 4: 1/3  [==>  4: 1/3]
 5: 3/2  [==>  5: 3/2]
 6: 2/3  [==>  6: 2/3]
 7: 3    [==>  7: 3  ]
 8: 1/4  [==>  8: 1/4]
 9: 4/3  [==>  9: 4/3]
10: 3/5  [==> 10: 3/5]
11: 5/2  [==> 11: 5/2]
12: 2/5  [==> 12: 2/5]
13: 5/3  [==> 13: 5/3]
14: 3/4  [==> 14: 3/4]
15: 4    [==> 15: 4  ]
16: 1/5  [==> 16: 1/5]
17: 5/4  [==> 17: 5/4]
18: 4/7  [==> 18: 4/7]
19: 7/3  [==> 19: 7/3]
20: 3/8  [==> 20: 3/8]

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}}==
<syntaxhighlight lang="python">from fractions import Fraction
from math import floor
from itertools import islice, groupby


def cw():
    a = Fraction(1)
    while True:
        yield a
        a = 1 / (2 * floor(a) + 1 - a)

def r2cf(rational):
    num, den = rational.numerator, rational.denominator
    while den:
        num, (digit, den) = den, divmod(num, den)
        yield digit

def get_term_num(rational):
    ans, dig, pwr = 0, 1, 0
    for n in r2cf(rational):
        for _ in range(n):
            ans |= dig << pwr
            pwr += 1
        dig ^= 1
    return ans

          
if __name__ == '__main__':
    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.")</syntaxhighlight>

{{out}}
<pre>TERMS 1..20:  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

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. 

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" line>my @calkin-wilf = Any, 1, {1 / (.Int × 2 + 1 - $_)} … *;

# Rational to Calkin-Wilf index
sub r2cw (Rat $rat) { :2( join '', flat (flat (1,0) xx *) Zxx reverse r2cf $rat ) }

# The task

say "First twenty terms of the Calkin-Wilf sequence: ",
    @calkin-wilf[1..20]».&prat.join: ', ';

say "\n99991st through 100000th: ",
    (my @tests = @calkin-wilf[99_991 .. 100_000])».&prat.join: ', ';

say "\nCheck reversibility: ", @tests».Rat».&r2cw.join: ', ';

say "\n83116/51639 is at index: ", r2cw 83116/51639;


# Helper subs
sub r2cf (Rat $rat is copy) { # Rational to continued fraction
    gather loop {
	    $rat -= take $rat.floor;
	    last if !$rat;
	    $rat = 1 / $rat;
    }
}

sub prat ($num) { # pretty Rat
    return $num unless $num ~~ Rat|FatRat;
    return $num.numerator if $num.denominator == 1;
    $num.nude.join: '/';
}</syntaxhighlight>
{{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

99991st through 100000th: 1085/303, 303/1036, 1036/733, 733/1163, 1163/430, 430/987, 987/557, 557/684, 684/127, 127/713

Check reversibility: 99991, 99992, 99993, 99994, 99995, 99996, 99997, 99998, 99999, 100000

83116/51639 is at index: 123456789</pre>
=={{header|REXX}}==
The meat of this REXX program was provided by Paul Kislanko.
<syntaxhighlight lang="rexx">/*REXX pgm finds the Nth value of the  Calkin─Wilf  sequence (which will be a fraction),*/
/*────────────────────── or finds which sequence number contains a specified fraction). */
numeric digits 2000                              /*be able to handle ginormic integers. */
parse arg LO HI te .                             /*obtain optional arguments from the CL*/
if LO=='' | LO==","   then LO=  1                /*Not specified?  Then use the default.*/
if HI=='' | HI==","   then HI= 20                /* "      "         "   "   "     "    */
if te=='' | te==","   then te= '/'               /* "      "         "   "   "     "    */
if datatype(LO, 'W')  then call CW_terms         /*Is LO numeric?  Then show some terms.*/
if pos('/', te)>0     then call CW_frac  te      /*Does TE have a / ?   Then find term #*/
exit 0
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg ?;  do jc=length(?)-3  to 1  by -3; ?=insert(',', ?, jc); end;  return ?
th:     parse arg th; return word('th st nd rd', 1+(th//10) *(th//100%10\==1) *(th//10<4))
/*──────────────────────────────────────────────────────────────────────────────────────*/
CW_frac:   procedure; parse arg p '/' q .;       say
           if q==''  then do;  p= 83116;         q= 51639;  end
           n= rle2dec( frac2cf(p q) );                    @CWS= 'the Calkin─Wilf sequence'
           say 'for '  p"/"q',  the element number for'   @CWS    "is: "    commas(n)th(n)
           if length(n)<10  then return
           say;  say 'The above number has '     commas(length(n))      " decimal digits."
           return
/*──────────────────────────────────────────────────────────────────────────────────────*/
CW_term:   procedure;  parse arg z;                 dd= 1;               nn= 0
                                       do z
                                       parse value  dd  dd*(2*(nn%dd)+1)-nn   with  nn  dd
                                       end   /*z*/
           return nn'/'dd
/*──────────────────────────────────────────────────────────────────────────────────────*/
CW_terms:  $=;        if LO\==0  then  do j=LO  to HI;   $= $  CW_term(j)','
                                       end   /*j*/
           if $==''  then return
           say 'Calkin─Wilf sequence terms for '  commas(LO)  " ──► "  commas(HI)  ' are:'
           say strip( strip($), 'T', ",")
           return
/*──────────────────────────────────────────────────────────────────────────────────────*/
frac2cf:   procedure;  parse arg p q;  if q==''  then return p;          cf= p % q;   m= q
           p= p - cf*q;                n= p;        if p==0  then return cf
                         do k=1  until n==0;        @.k= m % n
                         m= m  -  @.k * n;    parse value  n m   with   m n   /*swap N M*/
                         end   /*k*/
                                              /*for inverse Calkin─Wilf, K must be even.*/
           if k//2  then do;  @.k= @.k - 1;   k= k + 1;    @.k= 1;   end
                         do k=1  for k;       cf= cf @.k;            end  /*k*/
           return cf
/*──────────────────────────────────────────────────────────────────────────────────────*/
rle2dec:   procedure;  parse arg f1 rle;                       obin= copies(1, f1)
                               do until rle=='';               parse var rle f0 f1 rle
                               obin= copies(1, f1)copies(0, f0)obin
                               end   /*until*/
           return x2d( b2x(obin) )            /*RLE2DEC: Run Length Encoding ──► decimal*/</syntaxhighlight>
{{out|output|text=&nbsp; when using the default inputs:}}
<pre>
Calkin─Wilf sequence terms for  1  ──►  20  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

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}}==
<syntaxhighlight lang="rust">// [dependencies]
// num = "0.3"

use num::rational::Rational;

fn calkin_wilf_next(term: &Rational) -> Rational {
    Rational::from_integer(1) / (Rational::from_integer(2) * term.floor() + 1 - term)
}

fn continued_fraction(r: &Rational) -> Vec<isize> {
    let mut a = *r.numer();
    let mut b = *r.denom();
    let mut result = Vec::new();
    loop {
        let (q, r) = num::integer::div_rem(a, b);
        result.push(q);
        a = b;
        b = r;
        if a == 1 {
            break;
        }
    }
    let len = result.len();
    if len != 0 && len % 2 == 0 {
        result[len - 1] -= 1;
        result.push(1);
    }
    result
}

fn term_number(r: &Rational) -> usize {
    let mut result: usize = 0;
    let mut d: usize = 1;
    let mut p: usize = 0;
    for n in continued_fraction(r) {
        for _ in 0..n {
            result |= d << p;
            p += 1;
        }
        d ^= 1;
    }
    result
}

fn main() {
    println!("First 20 terms of the Calkin-Wilf sequence are:");
    let mut term = Rational::from_integer(1);
    for i in 1..=20 {
        println!("{:2}: {}", i, term);
        term = calkin_wilf_next(&term);
    }
    let r = Rational::new(83116, 51639);
    println!("{} is the {}th term of the sequence.", r, term_number(&r));
}</syntaxhighlight>

{{out}}
<pre>
First 20 terms of the Calkin-Wilf sequence are:
 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
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}}==
<syntaxhighlight lang="ruby">func calkin_wilf(n) is cached {
    return 1 if (n == 1)
    1/(2*floor(__FUNC__(n-1)) + 1 - __FUNC__(n-1))
}

func r2cw(r) {

    var cfrac = r.as_cfrac
    cfrac.len.is_odd || return nil

    Num(cfrac.flip.map_kv {|k,v| (k.is_odd ? '0' : '1') * v }.join, 2)
}

with (20) {|n|
    say "First #{n} terms of the Calkin-Wilf sequence:"
    say calkin_wilf.map(1..n)
}

with (83116/51639) {|r|
    say ("\n#{r.as_rat} is at index: ", r2cw(r))
}</syntaxhighlight>
{{out}}
<pre>
First 20 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]

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>

=={{header|Wren}}==
{{libheader|Wren-rat}}
{{libheader|Wren-fmt}}
<syntaxhighlight lang="wren">import "./rat" for Rat
import "./fmt" for Fmt, Conv

var calkinWilf = Fn.new { |n|
    var cw = List.filled(n, null)
    cw[0] = Rat.one
    for (i in 1...n) {
        var t = cw[i-1].floor * 2 - cw[i-1] + 1
        cw[i] = Rat.one / t
    }
    return cw
}

var toContinued = Fn.new { |r|
    var a = r.num
    var b = r.den
    var res = []
    while (true) {
        res.add((a/b).floor)
        var t = a % b
        a = b
        b = t
        if (a == 1) break
    }
    if (res.count%2 == 0) { // ensure always odd
        res[-1] = res[-1] - 1
        res.add(1)
    }
    return res
}

var getTermNumber = Fn.new { |cf|
    var b = ""
    var d = "1"
    for (n in cf) {
        b = (d * n) + b
        d = (d == "1") ? "0" : "1"
    }
    return Conv.atoi(b, 2)
}

var cw = calkinWilf.call(20)
System.print("The first 20 terms of the Calkin-Wilf sequence are:")
Rat.showAsInt = true
for (i in 1..20) Fmt.print("$2d: $s", i, cw[i-1])
System.print()
var r = Rat.new(83116, 51639)
var cf = toContinued.call(r)
var tn = getTermNumber.call(cf)
Fmt.print("$s is the $,r term of the sequence.", r, tn)</syntaxhighlight>

{{out}}
<pre>
The first 20 terms of the Calkin-Wilf sequence are:
 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

83116/51639 is the 123,456,789th term of the sequence.
</pre>