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``````