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Math::PlanePath::DiagonalsAlternating -- points in diagonal stripes of alternating directions
use Math::PlanePath::DiagonalsAlternating; my $path = Math::PlanePath::DiagonalsAlternating->new; my ($x, $y) = $path->n_to_xy (123);
This path follows successive diagonals going from the Y axis down to the X axis and then back again,
7 | 29 6 | 28 30 5 | 16 27 31 4 | 15 17 26 ... 3 | 7 14 18 25 2 | 6 8 13 19 24 1 | 2 5 9 12 20 23 Y=0 | 1 3 4 10 11 21 22 +---------------------------- X=0 1 2 3 4 5 6
The triangular numbers 1,3,6,10,etc k*(k+1)/2 are the start of each run up or down alternately on the X axis and Y axis. N=1,6,15,28,etc on the Y axis (Y even) are the hexagonal numbers j*(2j-1). N=3,10,21,36,etc on the X axis (X odd) are the hexagonal numbers of the second kind j*(2j+1).
The default is to number points starting N=1 as shown above. An optional
n_start can give a different start, in the same pattern. For example to start at 0,
n_start => 0 4 | 14 3 | 6 13 2 | 5 7 12 1 | 1 4 8 11 Y=0 | 0 2 3 9 10 +----------------- X=0 1 2 3 4
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::DiagonalsAlternating->new ()
$path = Math::PlanePath::DiagonalsAlternating->new (n_start => $n)
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number
$non the path.
$n < 1the return is an empty list, it being considered the path begins at 1.
Rectangle to N Range
Within each row increasing X is increasing N, and in each column increasing Y is increasing N. So in a rectangle the lower left corner is the minimum N and the upper right is the maximum N.
| N max | ----------+ | | ^ | | | | | | | ----> | | +---------- | N min +-------------------
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
n_start=1 A131179 N on X axis (extra initial 0) A128918 N on Y axis (extra initial 1) A001844 N on X=Y diagonal A038722 permutation N at transpose Y,X n_start=0 A003056 X+Y A004247 X*Y A049581 abs(X-Y) A048147 X^2+Y^2 A004198 X bit-and Y A003986 X bit-or Y A003987 X bit-xor Y A004197 min(X,Y) A003984 max(X,Y) A101080 HammingDist(X,Y) A023531 dSum = dX+dY, being 1 at N=triangular+1 (and 0) A046092 N on X=Y diagonal A061579 permutation N at transpose Y,X A056011 permutation N at points by Diagonals,direction=up order A056023 permutation N at points by Diagonals,direction=down runs alternately up and down, both are self-inverse
The coordinates such as A003056 X+Y are the same here as in the Diagonals path.
DiagonalsAlternating transposes X,Y -> Y,X in every second diagonal but forms such as X+Y are unchanged by swapping to Y+X.
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
This file is part of Math-PlanePath.
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