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

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).

   NStart
       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

Formulas

RectangletoNRange
       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
           +-------------------

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.

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/>.

perl v5.32.0                                       2021-01-23              Math::PlanePat...nalsAlternating(3pm)

Name

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

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
             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.

Synopsis

        use Math::PlanePath::DiagonalsAlternating;
        my $path = Math::PlanePath::DiagonalsAlternating->new;
        my ($x, $y) = $path->n_to_xy (123);