| Math::PlanePath::DiagonalsAlternating(3pm) | User Contributed Perl Documentation | Math::PlanePath::DiagonalsAlternating(3pm) |
Math::PlanePath::DiagonalsAlternating -- points in diagonal stripes of alternating directions
use Math::PlanePath::DiagonalsAlternating; my $path = Math::PlanePath::DiagonalsAlternating->new; my ($x, $y) = $path->n_to_xy (123);
This path follows successive diagonals going from the Y axis down to the X axis and then back again,
7 | 29
6 | 28 30
5 | 16 27 31
4 | 15 17 26 ...
3 | 7 14 18 25
2 | 6 8 13 19 24
1 | 2 5 9 12 20 23
Y=0 | 1 3 4 10 11 21 22
+----------------------------
X=0 1 2 3 4 5 6
The triangular numbers 1,3,6,10,etc k*(k+1)/2 are the start of each run up or down alternately on the X axis and Y axis. N=1,6,15,28,etc on the Y axis (Y even) are the hexagonal numbers j*(2j-1). N=3,10,21,36,etc on the X axis (X odd) are the hexagonal numbers of the second kind j*(2j+1).
The default is to number points starting N=1 as shown above. An optional "n_start" can give a different start, in the same pattern. For example to start at 0,
n_start => 0
4 | 14
3 | 6 13
2 | 5 7 12
1 | 1 4 8 11
Y=0 | 0 2 3 9 10
+-----------------
X=0 1 2 3 4
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
For "$n < 1" the return is an empty list, it being considered the path begins at 1.
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.
Math::PlanePath, Math::PlanePath::Diagonals, Math::PlanePath::DiagonalsOctant
<http://user42.tuxfamily.org/math-planepath/index.html>
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath 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.
Math-PlanePath 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 Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
| 2018-03-18 | perl v5.26.1 |