Reference

Math::PlanePath::StaircaseAlternating(3pm)

NAME

Math::PlanePath::StaircaseAlternating -- stair-step diagonals up and down

SYNOPSIS

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

DESCRIPTION

This path makes a staircase pattern up from Y axis down to the X and then back up again.

    10       46
              |
     9       47--48
                  |
     8       45  49--50
              |       |
     7       44--43  51--52
                  |       |
     6       16  42--41  53--54
              |       |       |
     5       17--18  40--39  55--...
                  |       |
     4       15  19--20  38--37
              |       |       |
     3       14--13  21--22  36--35
                  |       |       |
     2        2  12--11  23--24  34--33
              |       |       |       |
     1        3-- 4  10-- 9  25--26  32--31
                  |       |       |       |
    Y=0 ->    1   5-- 6   8-- 7  27--28  30--29
              ^
             X=0  1   2   3   4   5   6   7   8

Square Ends

Option "end_type => "square"" changes the path as follows, omitting one point at each end so as to square up the joins.

     9       42--43
              |   |
     8       41  44--45
              |       |
     7       40--39  46--47
                  |       |
     6        .  38--37  48--49
                      |       |
     5       14--15  36--35  50--...
              |   |       |
     4       13  16--17  34--33
              |       |       |
     3       12--11  18--19  32--31
                  |       |       |
     2        .  10-- 9  20--21  30--29
                      |       |       |
     1        2-- 3   8-- 7  22--23  28--27
              |   |       |       |       |
    Y=0 ->    1   4-- 5-- 6   .  24--25--26
              ^
             X=0  1   2   3   4   5   6   7   8

The effect is to shorten each diagonal by a constant 1 each time. The lengths of each diagonal still grow by +4 each time (or by +16 up and back).

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                  n_start => 0, end_type=>"square"
    46 47                            41 42
    44 48 49                         40 43 44
    43 42 50 51                      39 38 45 46
    15 41 40 52 53                      37 36 47 48
    16 17 39 38 ...                  13 14 35 34 ...
    14 18 19 37 36                   12 15 16 33 32
    13 12 20 21 35 34                11 10 17 18 31 30
     1 11 10 22 23 33 32                 9  8 19 20 29 28
     2  3  9  8 24 25 31 30           1  2  7  6 21 22 27 26
     0  4  5  7  6 26 27 29 28        0  3  4  5    23 24 25

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

"$path = Math::PlanePath::StaircaseAlternating->new ()"
"$path = Math::PlanePath::StaircaseAlternating->new (end_type => $str, n_start => $n)"

Create and return a new path object. The "end_type" choices are

    "jump"        (the default)
    "square"
"($x,$y) = $path->n_to_xy ($n)"

Return the X,Y coordinates of point number $n on the path.

OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

<http://oeis.org/A084849> (etc)

    end_type=jump, n_start=1  (the defaults)
      A084849    N on diagonal X=Y
    end_type=jump, n_start=0
      A014105    N on diagonal X=Y, second hexagonal numbers
    end_type=jump, n_start=2
      A096376    N on diagonal X=Y
    end_type=square, n_start=1
      A058331    N on diagonal X=Y, 2*squares+1
    end_type=square, n_start=0
      A001105    N on diagonal X=Y, 2*squares

SEE ALSO

Math::PlanePath, Math::PlanePath::Staircase, Math::PlanePath::DiagonalsAlternating

HOME PAGE

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

LICENSE

Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde

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/>.

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