/ 
Math-PlanePath-129
River stage one • 3 direct dependents • 4 total dependents
/ Math::PlanePath::StaircaseAlternating

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

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/>.