Math::PlanePath::DiagonalRationals -- rationals X/Y by diagonals
use Math::PlanePath::DiagonalRationals; my $path = Math::PlanePath::DiagonalRationals->new; my ($x, $y) = $path->n_to_xy (123);
This path enumerates positive rationals X/Y with no common factor, going in diagonal order from Y down to X.
17 | 96... 16 | 80 15 | 72 81 14 | 64 82 13 | 58 65 73 83 97 12 | 46 84 11 | 42 47 59 66 74 85 98 10 | 32 48 86 9 | 28 33 49 60 75 87 8 | 22 34 50 67 88 7 | 18 23 29 35 43 51 68 76 89 99 6 | 12 36 52 90 5 | 10 13 19 24 37 44 53 61 77 91 4 | 6 14 25 38 54 69 92 3 | 4 7 15 20 30 39 55 62 78 93 2 | 2 8 16 26 40 56 70 94 1 | 1 3 5 9 11 17 21 27 31 41 45 57 63 71 79 95 Y=0 | +--------------------------------------------------- X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
The order is the same as the Diagonals
path, but only those X,Y with no common factor are numbered.
1/1, N = 1 1/2, 1/2, N = 2 .. 3 1/3, 1/3, N = 4 .. 5 1/4, 2/3, 3/2, 4/1, N = 6 .. 9 1/5, 5/1, N = 10 .. 11
N=1,2,4,6,10,etc at the start of each diagonal (in the column at X=1) is the cumulative totient,
totient(i) = count numbers having no common factor with i i=K cumulative_totient(K) = sum totient(i) i=1
Option direction => 'up'
reverses the order within each diagonal to count upward from the X axis.
direction => "up" 8 | 27 7 | 21 26 6 | 17 5 | 11 16 20 25 4 | 9 15 24 3 | 5 8 14 19 2 | 3 7 13 23 1 | 1 2 4 6 10 12 18 22 Y=0| +--------------------------- X=0 1 2 3 4 5 6 7 8
The default is to number points starting N=1 as shown above. An optional n_start
can give a different start with the same shape, For example to start at 0,
n_start => 0 8 | 21 7 | 17 22 6 | 11 5 | 9 12 18 23 4 | 5 13 24 3 | 3 6 14 19 2 | 1 7 15 25 1 | 0 2 4 8 10 16 20 26 Y=0| +--------------------------- X=0 1 2 3 4 5 6 7 8
The diagonals are the same as the columns in CoprimeColumns
. For example the diagonal N=18 to N=21 from X=0,Y=8 down to X=8,Y=0 is the same as the CoprimeColumns
vertical at X=8. In general the correspondence is
Xdiag = Ycol Ydiag = Xcol - Ycol Xcol = Xdiag + Ydiag Ycol = Xdiag
CoprimeColumns
has an extra N=0 at X=1,Y=1 which is not present in DiagonalRationals
. (It would be Xdiag=1,Ydiag=0 which is 1/0.)
The points numbered or skipped in a column up to X=Y is the same as the points numbered or skipped on a diagonal, simply because X,Y no common factor is the same as Y,X+Y no common factor.
Taking the CoprimeColumns
as enumerating fractions F = Ycol/Xcol with 0 < F < 1 the corresponding diagonal rational 0 < R < infinity is
1 F R = ------- = --- 1/F - 1 1-F 1 R F = ------- = --- 1/R + 1 1+R
which is a one-to-one mapping between the fractions F < 1 and all rationals.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::DiagonalRationals->new ()
$path = Math::PlanePath::DiagonalRationals->new (direction => $str, n_start => $n)
Create and return a new path object. direction
(a string) can be
"down" (the default) "up"
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number $n
on the path. Points begin at 1 and if $n < 1
then the return is an empty list.
The current implementation is fairly slack and is slow on medium to large N. A table of cumulative totients is built and retained for the diagonal d=X+Y.
This enumeration of rationals is in Sloane's Online Encyclopedia of Integer Sequences in the following forms
http://oeis.org/A020652 (etc)
direction=down, n_start=1 (the defaults) A020652 X, numerator A020653 Y, denominator A038567 X+Y sum, starting from X=1,Y=1 A054431 by diagonals 1=coprime, 0=not (excluding X=0 row and Y=0 column) A054430 permutation N at Y/X reverse runs of totient(k) many integers A054424 permutation DiagonalRationals -> RationalsTree SB A054425 padded with 0s at non-coprimes A054426 inverse SB -> DiagonalRationals A060837 permutation DiagonalRationals -> FactorRationals direction=down, n_start=0 A157806 abs(X-Y) difference
direction=up swaps X,Y.
Math::PlanePath, Math::PlanePath::CoprimeColumns, Math::PlanePath::RationalsTree, Math::PlanePath::PythagoreanTree
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde
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