NAME

Math::PlanePath::Staircase -- integer points in stair-step diagonal stripes

SYNOPSIS

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

DESCRIPTION

This path makes a staircase pattern down from the Y axis to the X,

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

The 1,6,15,28,etc along the X axis at the end of each run are the hexagonal numbers k*(2*k-1). The diagonal 3,10,21,36,etc up from X=0,Y=1 is the second hexagonal numbers k*(2*k+1), formed by extending the hexagonal numbers to negative k. The two together are the triangular numbers k*(k+1)/2.

Legendre's prime generating polynomial 2*k^2+29 bounces around for some low values then makes a steep diagonal upwards from X=19,Y=1, at a slope 3 up for 1 across, but only 2 of each 3 drawn.

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
    28
    29 30
    15 31 32
    16 17 33 34
     6 18 19 35 36
     7  8 20 21 37 38
     1  9 10 22 23 ....
     2  3 11 12 24 25
     0  4  5 13 14 26 27

FUNCTIONS

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

"$path = Math::PlanePath::Staircase->new ()"
"$path = Math::PlanePath::AztecDiamondRings->new (n_start => $n)": Create and return a new staircase path object.
"$n = $path->xy_to_n ($x,$y)": Return the point number for coordinates "$x,$y". $x and $y are rounded to the nearest integers, which has the effect of treating each point $n as a square of side 1, so the quadrant x>=-0.5, y>=-0.5 is covered.
"($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)": The returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in the rectangle.

FORMULAS

Rectangle to N Range

Within each row increasing X is increasing N, and in each column increasing Y is increasing pairs of N. Thus for "rect_to_n_range()" the lower left corner vertical pair is the minimum N and the upper right vertical pair is the maximum N.

A given X,Y is the larger of a vertical pair when ((X^Y)&1)==1. If that happens at the lower left corner then it's X,Y+1 which is the smaller N, as long as Y+1 is in the rectangle. Conversely at the top right if ((X^Y)&1)==0 then it's X,Y-1 which is the bigger N, again as long as Y-1 is in the rectangle.

OEIS

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

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

    n_start=1 (the default)
      A084849    N on diagonal X=Y
      A210521    permutation N by diagonals, upwards
      A199855      inverse
    n_start=0
      A014105    N on diagonal X=Y, second hexagonal numbers
    n_start=2
      A128918    N on X axis, except initial 1,1
      A096376    N on diagonal X=Y

HOME PAGE

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

LICENSE

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/>.

2021-01-23

perl v5.32.0