--- name: "Math::PlanePath::Staircase" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::Staircase" --- # Math::PlanePath::Staircase(3pm) ## 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 \ (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 ``` ## SEE ALSO Math::PlanePath, Math::PlanePath::Diagonals, Math::PlanePath::Corner, Math::PlanePath::ToothpickSpiral ## HOME PAGE \ ## 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 \.