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Math-PlanePath-92
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::DiagonalsAlternating

NAME

Math::PlanePath::DiagonalsAlternating -- points in diagonal stripes of alternating directions

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, fall alternately on the X axis and Y axis. So 1,6,15,28,etc on the Y axis, and 3,10,21,36,etc on the X axis. Those on the Y axis are the hexagonal numbers j*(2j-1) and those on the X axis are the hexagonals 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 sequence. 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 ()

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.

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
  A061579    permutation N at transpose Y,X

SEE ALSO

Math::PlanePath, Math::PlanePath::Diagonals, Math::PlanePath::DiagonalsOctant

HOME PAGE

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE

Copyright 2010, 2011, 2012 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/>.