1Math::PlanePath::StaircUasseer(3C)ontributed Perl DocumeMnattaht:i:oPnlanePath::Staircase(3)
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6 Math::PlanePath::Staircase -- integer points in stair-step diagonal
7 stripes
8
10 use Math::PlanePath::Staircase;
11 my $path = Math::PlanePath::Staircase->new;
12 my ($x, $y) = $path->n_to_xy (123);
13
15 This path makes a staircase pattern down from the Y axis to the X,
16 8 29
18 |
19 7 30---31
20 |
21 6 16 32---33
22 | |
23 5 17---18 34---35
24 | |
25 4 7 19---20 36---37
26 | | |
27 3 8--- 9 21---22 38---39
28 | | |
29 2 2 10---11 23---24 40...
30 | | |
31 1 3--- 4 12---13 25---26
32 | | |
33 Y=0 -> 1 5--- 6 14---15 27---28
34 ^
36 X=0 1 2 3 4 5 6
37 The 1,6,15,28,etc along the X axis at the end of each run are the
39 hexagonal numbers k*(2*k-1). The diagonal 3,10,21,36,etc up from
40 X=0,Y=1 is the second hexagonal numbers k*(2*k+1), formed by extending
41 the hexagonal numbers to negative k. The two together are the
42 triangular numbers k*(k+1)/2.
43 Legendre's prime generating polynomial 2*k^2+29 bounces around for some
45 low values then makes a steep diagonal upwards from X=19,Y=1, at a
46 slope 3 up for 1 across, but only 2 of each 3 drawn.
47 N Start
49 The default is to number points starting N=1 as shown above. An
50 optional "n_start" can give a different start, in the same pattern.
51 For example to start at 0,
52 n_start => 0
54 28
56 29 30
57 15 31 32
58 16 17 33 34
59 6 18 19 35 36
60 7 8 20 21 37 38
61 1 9 10 22 23 ....
62 2 3 11 12 24 25
63 0 4 5 13 14 26 27
64
66 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
67 classes.
68 "$path = Math::PlanePath::Staircase->new ()"
70 "$path = Math::PlanePath::AztecDiamondRings->new (n_start => $n)"
71 Create and return a new staircase path object.
72 "$n = $path->xy_to_n ($x,$y)"
74 Return the point number for coordinates "$x,$y". $x and $y are
75 rounded to the nearest integers, which has the effect of treating
76 each point $n as a square of side 1, so the quadrant x>=-0.5,
77 y>=-0.5 is covered.
78 "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
80 The returned range is exact, meaning $n_lo and $n_hi are the
81 smallest and biggest in the rectangle.
82
84 Rectangle to N Range
85 Within each row increasing X is increasing N, and in each column
86 increasing Y is increasing pairs of N. Thus for rect_to_n_range() the
87 lower left corner vertical pair is the minimum N and the upper right
88 vertical pair is the maximum N.
89 A given X,Y is the larger of a vertical pair when ((X^Y)&1)==1. If
91 that happens at the lower left corner then it's X,Y+1 which is the
92 smaller N, as long as Y+1 is in the rectangle. Conversely at the top
93 right if ((X^Y)&1)==0 then it's X,Y-1 which is the bigger N, again as
94 long as Y-1 is in the rectangle.
95
97 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
98 this path include
99 <http://oeis.org/A084849> (etc)
101 n_start=1 (the default)
103 A084849 N on diagonal X=Y
104 A210521 permutation N by diagonals, upwards
105 A199855 inverse
106 n_start=0
108 A014105 N on diagonal X=Y, second hexagonal numbers
109 n_start=2
111 A128918 N on X axis, except initial 1,1
112 A096376 N on diagonal X=Y
113
115 Math::PlanePath, Math::PlanePath::Diagonals, Math::PlanePath::Corner,
116 Math::PlanePath::ToothpickSpiral
117
119 <http://user42.tuxfamily.org/math-planepath/index.html>
120
122 Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
123 2020, 2021 Kevin Ryde
124 This file is part of Math-PlanePath.
126 Math-PlanePath is free software; you can redistribute it and/or modify
128 it under the terms of the GNU General Public License as published by
129 the Free Software Foundation; either version 3, or (at your option) any
130 later version.
131 Math-PlanePath is distributed in the hope that it will be useful, but
133 WITHOUT ANY WARRANTY; without even the implied warranty of
134 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
135 General Public License for more details.
136 You should have received a copy of the GNU General Public License along
138 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
139perl v5.36.0 2023-01-20 Math::PlanePath::Staircase(3)