/ 
Math-PlanePath-129
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 up again,

  5  |  16
     |   |\
  4  |  15  17
     |    \   \
  3  |   7  14  18
     |   |\   \   \
  2  |   6   8  13  19  ...
     |    \   \   \   \   \
  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).

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

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
  A319572    X coordinate
  A319573    Y coordinate
  A319571    X,Y coordinates together
  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.

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, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 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/>.