Math::PlanePath::CfracDigits -- continued fraction terms encoded by digits
use Math::PlanePath::CfracDigits; my $path = Math::PlanePath::CfracDigits->new (tree_type => 'Kepler'); my ($x, $y) = $path->n_to_xy (123);
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.
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".
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.
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 ^^^^ ^^^^
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.
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)
Math::PlanePath, Math::PlanePath::FractionsTree, Math::PlanePath::CoprimeColumns
Math::PlanePath::RationalsTree, Math::PlanePath::GcdRationals, Math::PlanePath::DiagonalRationals
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2012, 2013, 2014 Kevin Ryde
This file is part of Math-PlanePath.
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