Math::...ircase(3pm)

💅 Math::PlanePath::Staircase 💅

hii bestie, welcome to Math::PlanePath::Staircase, straight outta the User Contributed Perl Documentation collection ✨ it's a Perl module, it draws a staircase, and honestly? it ate.

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✨ NAME ✨

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

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🌸 SYNOPSIS 🌸

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

like, that's it. that's the whole vibe. 💖

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🦋 DESCRIPTION 🦋

so this path does the thing where it makes a staircase pattern, going down from the Y axis over to the X axis. no cap, here's the actual picture:

     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

okay so the 1,6,15,28, etc that sit along the X axis at the end of each run? those are the hexagonal numbers k*(2*k-1) 🐝✨

and the diagonal 3,10,21,36, etc going up from X=0,Y=1 is the second hexagonal numbers k*(2*k+1) — which you get by extending the hexagonal numbers to negative k. the two of them together? that's the triangular numbers k*(k+1)/2. math is so giving honestly.

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

🎀 N Start 🎀

default is to number points starting at N=1, exactly like the pic above. but there's an optional "n_start" that lets you start somewhere else, same pattern, different numbers. like say you wanna 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

look at that. gorgeous. 🔥

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👑 FUNCTIONS 👑

for the stuff that every path class has in common, go check out "FUNCTIONS" in Math::PlanePath 💫

• "$path = Math::PlanePath::Staircase->new ()"
• "$path = Math::PlanePath::AztecDiamondRings->new (n_start => $n)" — makes a brand new staircase path object and hands it to you 🫶
• "$n = $path->xy_to_n ($x,$y)" — gives you back the point number for the coords "$x,$y". $x and $y get rounded to the nearest integers, which lowkey means every point $n is treated like a square of side 1, so the whole quadrant x>=-0.5, y>=-0.5 is covered 💀
• "($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 literally the smallest and biggest in the rectangle. no guessing, periodt ✨

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🧚 FORMULAS 🧚

🌈 Rectangle to N Range 🌈

within each row, increasing X means increasing N. and in each column, increasing Y means increasing pairs of N. so 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 that's the smaller N — as long as Y+1 is inside the rectangle. and conversely, at the top right, if ((X^Y)&1)==0 then it's X,Y-1 that's the bigger N, again as long as Y-1 is in the rectangle 🦋

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⭐ OEIS ⭐

entries in Sloane's Online Encyclopedia of Integer Sequences that relate 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

iykyk 😌

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💀 SEE ALSO 💀

• Math::PlanePath
• Math::PlanePath::Diagonals
• Math::PlanePath::Corner
• Math::PlanePath::ToothpickSpiral

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🏠 HOME PAGE 🏠

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

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📜 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 <http://www.gnu.org/licenses/>.

bestie pls actually read the license, it's giving legal realness 💅

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okay that's the whole page!! this one's perl v5.32.0, dated 2021-01-23, and it lives under the Math::...ircase(3pm) name — stay staircasing, queen 👑💖✨