1Math::NumSeq::PolygonalU(s3e)r Contributed Perl DocumentaMtaitohn::NumSeq::Polygonal(3)
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NAME

6       Math::NumSeq::Polygonal -- polygonal numbers, triangular, square,
7       pentagonal, etc
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SYNOPSIS

10        use Math::NumSeq::Polygonal;
11        my $seq = Math::NumSeq::Polygonal->new (polygonal => 7);
12        my ($i, $value) = $seq->next;
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DESCRIPTION

15       The sequence of polygonal numbers.  The 3-gonals are the triangular
16       numbers i*(i+1)/2, the 4-gonals are squares i*i, the 5-gonals are
17       pentagonals (3i-1)*i/2, etc.
18
19       In general the k-gonals for k>=3 are
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21           P(i) = (k-2)/2 * i*(i+1) - (k-3)*i
22
23       The values are how many points are in a triangle, square, pentagon,
24       hexagon, etc of side i.  For example the triangular numbers,
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26                                                d
27                                    c          c d
28                       b           b c        b c d
29           a          a b         a b c      a b c d
30
31           i=1        i=2         i=3        i=4
32           value=1    value=3     value=6    value=10
33
34       Or the squares,
35
36                                             d d d d
37                                  c c c      c c c d
38                      b b         b b c      b b c d
39           a          a b         a b c      a b c d
40
41           i=1        i=2         i=3        i=4
42           value=1    value=4     value=9    value=16
43
44       Or pentagons (which should be a pentagonal grid, so skewing a bit
45       here),
46
47                                                     d
48                                                   d   d
49                                      c          d  c    d
50                                    c   c      d  c   c    d
51                         b        c  b    c     c  b    c d
52                       b   b       b   b c       b   b c d
53           a            a b         a b c         a b c d
54
55           i=1        i=2         i=3          i=4
56           value=1    value=5     value=12     value=22
57
58       The letters "a", "b" "c" show the extra added onto the previous figure
59       to grow its points.  Each side except two are extended.  In general the
60       k-gonals increment by k-2 sides of i points, plus 1 at the end of the
61       last side, so
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63          P(i+1) = P(i) + (k-2)*i + 1
64
65   Second Kind
66       Option "pairs => 'second'" gives the polygonals of the second kind,
67       which are the same formula but with a negative i.
68
69           S(i) = P(-i) = (k-2)/2 * i*(i-1) + (k-3)*i
70
71       The result is still positive values, bigger than the plain P(i).  For
72       example the pentagonals are 0,1,5,12,22,etc and the second pentagonals
73       are 0,2,7,15,26,etc.
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75   Both Kinds
76       "pairs => 'both'" gives the firsts and seconds interleaved.  P(0) and
77       S(0) are both 0 and that value is given just once at i=0, so
78
79           0, P(1), S(1), P(2), S(2), P(3), S(3), ...
80
81   Average
82       Option "pairs => 'average'" is the average of the first and second,
83       which ends up being simply a multiple of the perfect squares,
84
85           A(i) = (P(i)+S(i))/2
86                = (k-2)/2 * i*i
87
88       This is an integer if k is even, or k odd and i is even.  If k and i
89       both odd then it's an 0.5 fraction.
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FUNCTIONS

92       See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence
93       classes.
94
95       "$seq = Math::NumSeq::Polygonal->new ()"
96       "$seq = Math::NumSeq::Polygonal->new (pairs => $str)"
97           Create and return a new sequence object.  The default is the
98           polygonals of the "first" kind, or the "pairs" option (a string)
99           can be
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101               "first"
102               "second"
103               "both"
104               "average"
105
106   Random Access
107       "$value = $seq->ith($i)"
108           Return the $i'th polygonal value, of the given "pairs" type.
109
110       "$bool = $seq->pred($value)"
111           Return true if $value is a polygonal number, of the given "pairs"
112           type.
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114       "$i = $seq->value_to_i_estimate($value)"
115           Return an estimate of the i corresponding to $value.
116

SEE ALSO

118       Math::NumSeq, Math::NumSeq::Cubes
119

HOME PAGE

121       <http://user42.tuxfamily.org/math-numseq/index.html>
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LICENSE

124       Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2018, 2019, 2020, 2021
125       Kevin Ryde
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127       Math-NumSeq is free software; you can redistribute it and/or modify it
128       under the terms of the GNU General Public License as published by the
129       Free Software Foundation; either version 3, or (at your option) any
130       later version.
131
132       Math-NumSeq 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
137       You should have received a copy of the GNU General Public License along
138       with Math-NumSeq.  If not, see <http://www.gnu.org/licenses/>.
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142perl v5.38.0                      2023-07-20        Math::NumSeq::Polygonal(3)
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