Calkin-Wilf sequence

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

It can be calculated recursively as follows:

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

Task part 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, ..., a_n]  =  [a₀; a₁, a₂ ,..., a_n-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.

Task part 2

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

See also

Wikipedia entry: Calkin-Wilf tree
Continued fraction
Continued fraction/Arithmetic/Construct from rational number

11l

Translation of: Nim

<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.’)</lang>

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

<lang applescript>-- Return the first n terms of the sequence. Tree generation. Faster for this purpose. on CalkinWilfSequence(n)

   script o
       property sequence : Template: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</lang>

Output:

<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"</lang>

Arturo

<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."]</lang>

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.

<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⟩</lang> <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 ⟩ ⟩</lang>

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

<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) );</lang>

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

<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";

}</lang>

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

<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] </lang>

Output:

Find index of a given term

Translation of: Pascal

<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]

</lang>

Output:

83116/51639 IS AT 123456789

F#

The Function

<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 </lang>

The Tasks

first 20

<lang fsharp> cW |> Seq.take 20 |> Seq.iter(fun(n,g)->printf "%d/%d " n g);printfn "" </lang>

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

<lang fsharp> printfn "%d" (1+Seq.findIndex(fun n->n=(83116,51639)) cW) </lang>

Output:

123456789

Factor

Works with: Factor version 0.99 2020-08-14

<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</lang>

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.

Forth

Works with: gforth version 0.7.3

<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</lang>

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 
83116 / 51639 123456789  ok

FreeBASIC

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

<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."</lang>

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. <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))

}</lang>

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.

Haskell

<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)</lang>

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 <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 </lang>

Julia

Translation of: Wren

<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.")

</lang>

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

<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 </lang>

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. <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 </lang>

Output:

9/4<-35

Mathematica / Wolfram Language

<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[];
 ]
]</lang>

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.

<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."</lang>

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. <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. </lang>

Output:

<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. </lang>

Output:

Index of 83116/51639 is 123456789

Perl

Translation of: Raku

Library: ntheory

<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);</lang>

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

<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), !. </lang>

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

<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.")</lang>

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.

Quackery

cf is defined at Continued fraction/Arithmetic/Construct from rational number#Quackery.

<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</lang>

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 

123456789

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.

<lang perl6>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: '/';

}</lang>

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. <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*/</lang>

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

<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) </lang>

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

<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));

}</lang>

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

<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))

}</lang>

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

Vlang

Translation of: Go

s.

<lang 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.")

}</lang>

Output:

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.

Wren

Library: Wren-rat

Library: Wren-fmt

<lang ecmascript>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)</lang>

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.