NAME
Math::PlanePath::DiagonalRationals  rationals X/Y by diagonals
SYNOPSIS
use Math::PlanePath::DiagonalRationals;
my $path = Math::PlanePath::DiagonalRationals>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
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
Direction Up
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
N Start
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
Coprime Columns
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 1F
1 R
F =  = 
1/R + 1 1+R
which is a onetoone mapping between the fractions F < 1 and all rationals.
FUNCTIONS
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.
BUGS
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.
OEIS
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 noncoprimes
A054426 inverse SB > DiagonalRationals
A060837 permutation DiagonalRationals > FactorRationals
direction=down, n_start=0
A157806 abs(XY) difference
direction=up swaps X,Y.
SEE ALSO
Math::PlanePath, Math::PlanePath::CoprimeColumns, Math::PlanePath::RationalsTree, Math::PlanePath::PythagoreanTree
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.