SYNOPSIS
use Math::PlanePath::DiagonalsAlternating;
my $path = Math::PlanePath::DiagonalsAlternating>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path follows successive diagonals going from the Y axis down to the X axis and then back again,
7  29 6  28 30 5  16 27 31 4  15 17 26 ... 3  7 14 18 25 2  6 8 13 19 24 1  2 5 9 12 20 23 Y=0  1 3 4 10 11 21 22 + X=0 1 2 3 4 5 6
The triangular numbers 1,3,6,10,etc k*(k+1)/2 are the start of each run up or down alternately on the X axis and Y axis. N=1,6,15,28,etc on the Y axis (Y even) are the hexagonal numbers j*(2j1). N=3,10,21,36,etc on the X axis (X odd) are the hexagonal numbers of the second kind j*(2j+1).
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can give a different start, in the same pattern. For example to start at 0,
n_start => 0 4  14 3  6 13 2  5 7 12 1  1 4 8 11 Y=0  0 2 3 9 10 + X=0 1 2 3 4
FUNCTIONS
See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes. "$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 $n on the path.
For "$n < 1" the return is an empty list, it being considered the path begins at 1.
FORMULAS
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 +
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
 <http://oeis.org/A131179> (etc)
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(XY) A048147 X^2+Y^2 A004198 X bitand Y A003986 X bitor Y A003987 X bitxor 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 selfinverse
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.
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeThis file is part of MathPlanePath.
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/>.