Math::PlanePath::DiagonalsAlternating(3pm)

1Math::PlanePath::DiagonUaslesrAlCtoenrtnraitbiuntge(d3M)Paetrhl::DPolcaunmeePnattaht:i:oDniagonalsAlternating(3)
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NAME

6       Math::PlanePath::DiagonalsAlternating -- points in diagonal stripes of
7       alternating directions
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SYNOPSIS

10        use Math::PlanePath::DiagonalsAlternating;
11        my $path = Math::PlanePath::DiagonalsAlternating->new;
12        my ($x, $y) = $path->n_to_xy (123);
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DESCRIPTION

15       This path follows successive diagonals going from the Y axis down to
16       the X axis and then back up again,
17
18             5  |  16
19                |   |\
20             4  |  15  17
21                |    \   \
22             3  |   7  14  18
23                |   |\   \   \
24             2  |   6   8  13  19  ...
25                |    \   \   \   \   \
26             1  |   2   5   9  12  20  23
27                |   |\   \   \   \   \   \
28           Y=0  |   1   3-- 4  10--11  21--22
29                +----------------------------
30                  X=0   1   2   3   4   5   6
31
32       The triangular numbers 1,3,6,10,etc k*(k+1)/2 are the start of each run
33       up or down alternately on the X axis and Y axis.  N=1,6,15,28,etc on
34       the Y axis (Y even) are the hexagonal numbers j*(2j-1).
35       N=3,10,21,36,etc on the X axis (X odd) are the hexagonal numbers of the
36       second kind j*(2j+1).
37
38   N Start
39       The default is to number points starting N=1 as shown above.  An
40       optional "n_start" can give a different start, in the same pattern.
41       For example to start at 0,
42
43           n_start => 0
44
45             4  |  14
46             3  |   6 13
47             2  |   5  7 12
48             1  |   1  4  8 11
49           Y=0  |   0  2  3  9 10
50                +-----------------
51                  X=0  1  2  3  4
52

FUNCTIONS

54       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
55       classes.
56
57       "$path = Math::PlanePath::DiagonalsAlternating->new ()"
58       "$path = Math::PlanePath::DiagonalsAlternating->new (n_start => $n)"
59           Create and return a new path object.
60
61       "($x,$y) = $path->n_to_xy ($n)"
62           Return the X,Y coordinates of point number $n on the path.
63
64           For "$n < 1" the return is an empty list, it being considered the
65           path begins at 1.
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FORMULAS

68   Rectangle to N Range
69       Within each row increasing X is increasing N, and in each column
70       increasing Y is increasing N.  So in a rectangle the lower left corner
71       is the minimum N and the upper right is the maximum N.
72
73           |               N max
74           |     ----------+
75           |    |  ^       |
76           |    |  |       |
77           |    |   ---->  |
78           |    +----------
79           |   N min
80           +-------------------
81

OEIS

83       Entries in Sloane's Online Encyclopedia of Integer Sequences related to
84       this path include
85
86           <http://oeis.org/A131179> (etc)
87
88           n_start=1
89             A131179    N on X axis (extra initial 0)
90             A128918    N on Y axis (extra initial 1)
91             A001844    N on X=Y diagonal
92             A038722    permutation N at transpose Y,X
93
94           n_start=0
95             A319572    X coordinate
96             A319573    Y coordinate
97             A319571    X,Y coordinates together
98             A003056    X+Y
99             A004247    X*Y
100             A049581    abs(X-Y)
101             A048147    X^2+Y^2
102             A004198    X bit-and Y
103             A003986    X bit-or Y
104             A003987    X bit-xor Y
105             A004197    min(X,Y)
106             A003984    max(X,Y)
107             A101080    HammingDist(X,Y)
108             A023531    dSum = dX+dY, being 1 at N=triangular+1 (and 0)
109             A046092    N on X=Y diagonal
110             A061579    permutation N at transpose Y,X
111
112             A056011    permutation N at points by Diagonals,direction=up order
113             A056023    permutation N at points by Diagonals,direction=down
114                runs alternately up and down, both are self-inverse
115
116       The coordinates such as A003056 X+Y are the same here as in the
117       Diagonals path.  "DiagonalsAlternating" transposes X,Y -> Y,X in every
118       second diagonal but forms such as X+Y are unchanged by swapping to Y+X.
119

HOME PAGE

125       <http://user42.tuxfamily.org/math-planepath/index.html>
126

LICENSE

128       Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
129       2020, 2021 Kevin Ryde
130
131       This file is part of Math-PlanePath.
132
133       Math-PlanePath is free software; you can redistribute it and/or modify
134       it under the terms of the GNU General Public License as published by
135       the Free Software Foundation; either version 3, or (at your option) any
136       later version.
137
138       Math-PlanePath is distributed in the hope that it will be useful, but
139       WITHOUT ANY WARRANTY; without even the implied warranty of
140       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
141       General Public License for more details.
142
143       You should have received a copy of the GNU General Public License along
144       with Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.
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148perl v5.34.0                      2022-M0a1t-h2:1:PlanePath::DiagonalsAlternating(3)