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# Calkin-Wilf sequence

Calkin-Wilf sequence
You are encouraged to solve this task according to the task description, using any language you may know.

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

It can be calculated recursively as follows:

```       a1   =  1
an+1  =  1/(2⌊an⌋+1-an) for n > 1
```

• Show on this page terms 1 through 20 of the Calkin-Wilf sequence.

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:

```       [a0; a1, a2, ..., an]  =  [a0; a1, a2 ,..., an-1, 1]
```

Example

The fraction   9/4   has odd continued fraction representation     2; 3, 1,     giving a binary representation of   100011,
which means   9/4   appears as the   35th   term of the sequence.

• Find the position of the number   83116/51639   in the Calkin-Wilf sequence.

## 11l

Translation of: Nim
`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] seqL 20   V (n, d) = cw()   seq.append(I d == 1 {String(n)} E n‘/’d)print(seq.join(‘, ’)) cw = CalkinWilf()V index = 1L cw() != (83116, 51639)   index++print("\nThe element 83116/51639 is at position "index‘ in the sequence.’)`
Output:
```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.
```

## 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 sequenceend 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 sequenceend 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 positionend CalkinWilfSequencePosition -- Task code:local sequenceResult1, sequenceResult2, positionResult, output, astidset sequenceResult1 to CalkinWilfSequence(20)set sequenceResult2 to CalkinWilfSequence2(1, 20)set positionResult to CalkinWilfSequencePosition(83116, 51639)set astid to AppleScript's text item delimitersset 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 astidset output to output & (linefeed & "83116/51639 is term number " & positionResult)return output`
Output:
`"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, 3/8Ditto 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, 3/883116/51639 is term number 123456789"`

## Arturo

`n: new 1d: new 1calkinWilf: 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 => calkinWilfprint "The first 20 terms of the Calkwin-Wilf sequence are:"print map first20 'f -> ~"|f\0|/|f\1|" n: new 1d: new 1indx: 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."]`
Output:
```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.```

## BQN

BQN does not have rational number arithmetic yet, so it is manually implemented.

Part 2 runs in ~150 secs on CBQN.

`GCD` and `_while_` are idioms from BQNcrate.

`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 termCal ← {Next⍟𝕩 1‿1} •Show Cal 1+↕20 •Show {  cnt‿fr:  ⟨cnt+1,Next fr⟩} _while_ {  cnt‿fr:  fr ≢ 83116‿51639} ⟨1,1‿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 ⟩ ⟨ 8 5 ⟩ ⟩⟨ 123456789 ⟨ 83116 51639 ⟩ ⟩`

You can try Part 1 here. Second part can and will hang your browser, so it is best to try locally on CBQN.

## Bracmat

Translation of: Python
`( 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));`
Output:
```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```

## C++

Library: Boost
`#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";}`
Output:
```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.
```

## EDSAC order code

### Find first n terms

Translation of: Pascal
` [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][email protected]@[email protected]@AFT5FT4FH38#@[email protected][email protected]@[email protected][email protected]@[email protected]@[email protected] [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]         [email protected] TF [email protected] G56F [email protected] [print a followed by slash]         [email protected] TF [email protected] G56F [email protected] [email protected] [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] `
Output:
```    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
```

### Find index of a given term

Translation of: Pascal
` [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][email protected]@[email protected]@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 [email protected]@[email protected]@[email protected]@[email protected]#@[email protected][email protected]@[email protected]@[email protected]@[email protected][email protected]@[email protected]@[email protected]@[email protected] [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 ']            [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]            O   10 @ [set teleprinter to figures]            A    6#@ [acc := calculated index]            T      D [send to print subroutine]     [70]   A   70 @            G   56 F     [72]   [email protected] [email protected] [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]           [email protected] [email protected] [email protected] [email protected] [email protected] [print'ERROR']            G   72 @ [jump to common exit]            E   19 Z  [relative address of entry point]            P      F  [enter with acc = 0] `
Output:
```83116/51639 IS AT 123456789
```

## F#

### The Function

` // Calkin Wilf Sequence. Nigel Galloway: January 9th., 2021let 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 `

first 20
` cW |> Seq.take 20 |> Seq.iter(fun(n,g)->printf "%d/%d " n g);printfn "" `
Output:
```1/1 1/2 2/1 1/3 3/2 2/3 3/1 1/4 4/3 3/5 5/2 2/5 5/3 3/4 4/1 1/5 5/4 4/7 7/3 3/8
```
Indexof 83116/51639
` printfn "%d" (1+Seq.findIndex(fun n->n=(83116,51639)) cW) `
Output:
```123456789
```

## Factor

Works with: Factor version 0.99 2020-08-14
`USING: formatting io kernel lists lists.lazy mathmath.continued-fractions math.functions math.parser prettyprintsequences 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:" print20 calkin-wilf ltake [ pprint bl ] leach nl nl 83116/51639 cw-index "83116/51639 is at index %d.\n" printf`
Output:
```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.
```

## FreeBASIC

Uses the code from Greatest common divisor#FreeBASIC as an include.

`#include "gcd.bas" type rational    num as integer    den as integerend type dim shared as rational ONE, TWOONE.num = 1 : ONE.den = 1TWO.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 aend 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 retend function function cw_nextterm( q as rational ) as rational    dim as rational ret = (TWO*floor(q))    ret = ret + ONE : ret = ret - q     return ONE / retend 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 retend 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 qq.num = 1q.den = 1for i as integer = 1 to 20    print i, disp_rational(q)    q = cw_nextterm(q)next i q.num = 83116q.den = 51639print disp_rational(q)+" is the "+str(frac_to_int(q))+"th term."`
Output:
``` 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.```

## Fōrmulæ

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.

Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.

In this page you can see the program(s) related to this task and their results.

## Go

Translation of: 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.

`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))}`
Output:
```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.
```

`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 -> IntegercalkinWilfIdx = rld . cfo -- A continued fraction representation of a given rational number, guaranteed-- to have an odd length.cfo :: Rational -> NonEmpty Intcfo = oddLen . cf -- The canonical (i.e. shortest) continued fraction representation of a given-- rational number.cf :: Rational -> NonEmpty Intcf = 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 IntoddLen = 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 -> Integerrld = 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)`
Output:
``` 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.
```

## J

```   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
```

given definitions

` 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=: #[email protected]|[email protected](# 1 0 \$~ #)@[email protected] `

## Julia

Translation of: Wren
`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 resend 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 // 51639const cf = continued(r)const tn = term_number(cf)println("\$r is the \$tn-th term of the sequence.") `
Output:
```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.
```

## 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

Translation of: Pascal
` // 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 := 1next_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 pwr2loop2    STA nrDigits         LDA i         SUB pwr2         SUB pwr2         BRP double         BRA part2    // jump out if next power of 2 would exceed idouble   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 useddec_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 + bdigit_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_iexit     HLT// Constantsc1       DAT 1asc_hy   DAT 45asc_gt   DAT 62asc_sl   DAT 47// Variablesi        DATmax_i    DATpwr2     DATnrDigits DATdiff     DATa        DATb        DAT// end `
Output:
```  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
```

### Find index of a given term

Translation of: 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.

` // 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 indexloop     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 1a_gt_b   STA a         LDA index         ADD pwr2         STA index// In either case, on to next power of 2double   LDA pwr2         ADD pwr2         STA pwr2         BRA loop// Out of loop, add leading binary digit 1break    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 lineexit     HLT// Constantsc0       DAT 0c1       DAT 1asc_sl   DAT 47asc_lt   DAT 60asc_hy   DAT 45// Variablesnum      DATden      DATa        DATb        DATpwr2     DATindex    DAT// end `
Output:
```9/4<-35
```

## Mathematica / Wolfram Language

`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[];  ] ]`
Output:
```{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```

## 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.

`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."`
Output:
```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.```

## 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.

` 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 termsAlgorithm 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. `
Output:
```       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
```
` 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 unsigedbegin  // 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. `
Output:
```Index of 83116/51639 is 123456789
```

## Perl

Translation of: Raku
Library: ntheory
`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);`
Output:
```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```

## Phix

```with javascript_semantics
requires("1.0.0")   -- (new even() builtin)

function calkin_wilf(integer len)
sequence cw = repeat(0,len)
integer n=0, d=1
for i=1 to len do
{n,d} = {d,(floor(n/d)*2+1)*d-n}
cw[i] = {n,d}
end for
return cw
end function

function odd_length(sequence cf)
-- replace even length continued fraction with odd length equivalent
--  if remainder(length(cf),2)=0 then
if even(length(cf)) then
cf[\$] -= 1
cf &= 1
end if
return cf
end function

function to_continued_fraction(sequence r)
integer {a,b} = r
sequence cf = {}
while true do
cf &= floor(a/b)
{a, b} = {b, remainder(a,b)}
if a=1 then exit end if
end while
cf = odd_length(cf)
return cf
end function

function get_term_number(sequence cf)
sequence b = {}
integer d = 1
for i=1 to length(cf) do
b &= repeat(d,cf[i])
d = 1-d
end for
integer t = bits_to_int(b)
return t
end function

-- additional verification methods (2 of)
function i_to_cf(integer i)
--  sequence b = trim_tail(int_to_bits(i,32),0)&2,
sequence b = int_to_bits(i)&2,
cf = iff(b[1]=0?{0}:{})
while length(b)>1 do
for j=2 to length(b) do
if b[j]!=b[1] then
cf &= j-1
b = b[j..\$]
exit
end if
end for
end while
cf = odd_length(cf)
return cf
end function

function cf2r(sequence cf)
integer n=0, d=1
for i=length(cf) to 2 by -1 do
{n,d} = {d,n+d*cf[i]}
end for
return {n+cf[1]*d,d}
end function

function prettyr(sequence r)
integer {n,d} = r
return iff(d=1?sprintf("%d",n):sprintf("%d/%d",{n,d}))
end function

sequence cw = calkin_wilf(20)
printf(1,"The first 20 terms of the Calkin-Wilf sequence are:\n")
for i=1 to 20 do
string s = prettyr(cw[i]),
r = prettyr(cf2r(i_to_cf(i)))
integer t = get_term_number(to_continued_fraction(cw[i]))
printf(1,"%2d: %-4s [==> %2d: %-3s]\n", {i, s, t, r})
end for
printf(1,"\n")
sequence r = {83116,51639}
sequence cf = to_continued_fraction(r)
integer tn = get_term_number(cf)
printf(1,"%d/%d is the %,d%s term of the sequence.\n", r&{tn,ord(tn)})
```
Output:
```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.
```

## 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), !. `
Output:
```?- 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.
```

## Python

`from fractions import Fractionfrom math import floorfrom 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.")`
Output:
```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.```

## 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.

`my @calkin-wilf = Any, 1, {1 / (.Int × 2 + 1 - \$_)} … *; # Rational to Calkin-Wilf indexsub 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 subssub 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: '/';}`
Output:
```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```

## REXX

The meat of this REXX program was provided by Paul Kislanko.

`/*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*/`
output   when using the default inputs:
```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
```

## Ruby

Translation of: Python
`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  resend puts  cw.take(20).join(", ")puts  term_num  (83116/51639r) `
```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
```

## 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));}`
Output:
```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.
```

## Sidef

`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))}`
Output:
```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
```

## Wren

Library: Wren-rat
Library: Wren-fmt
`import "/rat" for Ratimport "/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 = truefor (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)`
Output:
```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.
```