Provided by: libmath-planepath-perl_129-1_all bug

NAME

       Math::PlanePath::CfracDigits -- continued fraction terms encoded by digits

SYNOPSIS

        use Math::PlanePath::CfracDigits;
        my $path = Math::PlanePath::CfracDigits->new (tree_type => 'Kepler');
        my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

       This path enumerates reduced fractions 0 < X/Y < 1 with X,Y no common factor using a
       method by Jeffrey Shallit encoding continued fraction terms in digit strings, as per

           Jeffrey Shallit, "Number Theory and Formal Languages", part 3,
           <https://cs.uwaterloo.ca/~shallit/Papers/ntfl.ps>

       Fractions up to a given denominator are covered by roughly N=den^2.28.  This is a much
       smaller N range than the run-length encoding in "RationalsTree" and "FractionsTree" (but
       is more than "GcdRationals").

           15  |    25  27      91          61 115         307     105 104
           14  |    23      48      65             119     111     103
           13  |    22  24  46  29  66  59 113 120 101 109  99  98
           12  |    17              60     114              97
           11  |    16  18  30  64  58 112 118 102  96  95
           10  |    14      28             100      94
            9  |    13  15      20  38      36  35
            8  |     8      21      39      34
            7  |     7   9  19  37  33  32
            6  |     5              31
            5  |     4   6  12  11
            4  |     2      10
            3  |     1   3
            2  |     0
            1  |
           Y=0 |
                ----------------------------------------------------------
               X=0   1   2   3   4   5   6   7   8   9  10  11  12  13  14

       A fraction 0 < X/Y < 1 has a finite continued fraction of the form

                             1
           X/Y = 0 + ---------------------
                                   1
                     q[1] + -----------------
                                         1
                            q[2] + ------------
                                ....
                                             1
                                   q[k-1] + ----
                                            q[k]

           where each  q[i] >= 1
           except last q[k] >= 2

       The terms are collected up as a sequence of integers >=0 by subtracting 1 from each and 2
       from the last.

           # each >= 0
           q[1]-1,  q[2]-1, ..., q[k-2]-1, q[k-1]-1, q[k]-2

       These integers are written in base-2 using digits 1,2.  A digit 3 is written between each
       term as a separator.

           base2(q[1]-1), 3, base2(q[2]-1), 3, ..., 3, base2(q[k]-2)

       If a term q[i]-1 is zero then its base-2 form is empty and there's adjacent 3s in that
       case.  If the high q[1]-1 is zero then a bare high 3, and if the last q[k]-2 is zero then
       a bare final 3.  If there's just a single term q[1] and q[1]-2=0 then the string is
       completely empty.  This occurs for X/Y=1/2.

       The resulting string of 1s,2s,3s is reckoned as a base-3 value with digits 1,2,3 and the
       result is N.  All possible strings of 1s,2s,3s occur (including the empty string) and so
       all integers N>=0 correspond one-to-one with an X/Y fraction with no common factor.

       Digits 1,2 in base-2 means writing an integer in the form

           d[k]*2^k + d[k-1]*2^(k-1) + ... + d[2]*2^2 + d[1]*2 + d[0]
           where each digit d[i]=1 or 2

       Similarly digits 1,2,3 in base-3 which is used for N,

           d[k]*3^k + d[k-1]*3^(k-1) + ... + d[2]*3^2 + d[1]*3 + d[0]
           where each digit d[i]=1, 2 or 3

       This is not the same as the conventional binary and ternary radix representations by
       digits 0,1 or 0,1,2 (ie. 0 to radix-1).  The effect of digits 1 to R is to change any 0
       digit to instead R and decrement the value above that position to compensate.

   Axis Values
       N=0,1,2,4,5,7,etc in the X=1 column is integers with no digit 0s in ternary.  N=0 is
       considered no digits at all and so no digit 0.  These points are fractions 1/Y which are a
       single term q[1]=Y-1 and hence no "3" separators, only a run of digits 1,2.  These N
       values are also those which are the same when written in digits 0,1,2 as when written in
       digits 1,2,3, since there's no 0s or 3s.

       N=0,3,10,11,31,etc along the diagonal Y=X+1 are integers which are ternary "10www..."
       where the w's are digits 1 or 2, so no digit 0s except the initial "10".  These points
       Y=X+1 points are X/(X+1) with continued fraction

                            1
           X/(X+1) =  0 + -------
                               1
                          1 + ---
                               X

       so q0=1 and q1=X, giving N="3,X-1" in digits 1,2,3, which is N="1,0,X-1" in normal
       ternary.  For example N=34 is ternary "1021" which is leading "10" and then X-1=7 ternary
       "21".

   Radix
       The optional "radix" parameter can select another base for the continued fraction terms,
       and corresponding radix+1 for the resulting N.  The default is radix=2 as described above.
       Any integer radix>=1 can be selected.  For example,

           radix => 5

           11  |    10   30  114  469   75  255 1549 1374  240  225
           10  |     9       109                1369       224
            9  |     8   24        74  254       234  223
            8  |     7        78       258        41
            7  |     5   18   73  253  228   40
            6  |     4                  39
            5  |     3   12   42   38
            4  |     2        37
            3  |     1    6
            2  |     0
            1  |
           Y=0 |
                ----------------------------------------------------
               X=0   1    2    3    4    5    6    7    8    9   10

       The X=1 column is integers with no digit 0 in base radix+1, so in radix=5 means no 0 digit
       in base-6.

   Radix 1
       The radix=1 case encodes continued fraction terms using only digit 1, which means runs of
       q many "1"s to add up to q, and then digit "2" as separator.

           N =  11111 2 1111 2 ... 2 1111 2 11111     base2 digits 1,2
                \---/   \--/         \--/   \---/
                q[1]-1  q[2]-1     q[k-1]-1 q[k]-2

       which becomes in plain binary

           N = 100000  10000   ...  10000  011111     base2 digits 0,1
               \----/  \---/        \---/  \----/
                q[1]    q[2]       q[k-1]  q[k]-1

       Each "2" becomes "0" in plain binary and carry +1 into the run of 1s above it.  That carry
       propagates through those 1s, turning them into 0s, and stops at the "0" above them (which
       had been a "2").  The low run of 1s from q[k]-2 has no "2" below it and is therefore
       unchanged.

           radix => 1

           11  |   511  32  18  21  39  55  29  26  48 767
           10  |   255      17              25     383
            9  |   127  16      19  27      24 191
            8  |    63      10      14      95
            7  |    31   8   9  13  12  47
            6  |    15              23
            5  |     7   4   6  11
            4  |     3       5
            3  |     1   2
            2  |     0
            1  |
           Y=0 |
                -------------------------------------------
               X=0   1   2   3   4   5   6   7   8   9  10

       The result is similar to "HCS Continued Fraction" in Math::PlanePath::RationalsTree.  But
       the lowest run is "0111" here, instead of "1000" as it is in the HCS.  So N-1 here, and a
       flip (Y-X)/X to map from X/Y<1 here to instead all rationals for the HCS tree.  For
       example

           CfracDigits radix=1       RationalsTree tree_type=HCS

           X/Y = 5/6                 (Y-X)/X = 1/5
           is at                     is at
           N = 23 = 0b10111          N = 24 = 0b11000
                       ^^^^                      ^^^^

FUNCTIONS

       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

       "$path = Math::PlanePath::CfracDigits->new ()"
       "$path = Math::PlanePath::CfracDigits->new (radix => $radix)"
           Create and return a new path object.

       "$n = $path->n_start()"
           Return 0, the first N in the path.

OEIS

       Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

           <http://oeis.org/A032924> (etc)

           radix=1
             A071766    X coordinate (numerator), except extra initial 1

           radix=2 (the default)
             A032924    N in X=1 column, ternary no digit 0 (but lacking N=0)

           radix=3
             A023705    N in X=1 column, base-4 no digit 0 (but lacking N=0)

           radix=4
             A023721    N in X=1 column, base-5 no digit 0 (but lacking N=0)

           radix=10
             A052382    N in X=1 column, decimal no digit 0 (but lacking N=0)

SEE ALSO

       Math::PlanePath, Math::PlanePath::FractionsTree, Math::PlanePath::CoprimeColumns

       Math::PlanePath::RationalsTree, Math::PlanePath::GcdRationals,
       Math::PlanePath::DiagonalRationals

       Math::ContinuedFraction

HOME PAGE

       <http://user42.tuxfamily.org/math-planepath/index.html>

LICENSE

       Copyright 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde

       This file is part of Math-PlanePath.

       Math-PlanePath is free software; you can redistribute it and/or modify it under the terms
       of the GNU General Public License as published by the Free Software Foundation; either
       version 3, or (at your option) any later version.

       Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY
       WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
       PURPOSE.  See the GNU General Public License for more details.

       You should have received a copy of the GNU General Public License along with Math-
       PlanePath.  If not, see <http://www.gnu.org/licenses/>.