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 => $str)"
97           Create and return a new sequence object.  The default is the
98           polygonals of the "first" kind, or the "pairs" option (a string)
99           can be
100
101               "first"
102               "second"
103               "both"
104               "average"
105
106   Random Access
107       "$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 Kevin Ryde
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126       Math-NumSeq is free software; you can redistribute it and/or modify it
127       under the terms of the GNU General Public License as published by the
128       Free Software Foundation; either version 3, or (at your option) any
129       later version.
130
131       Math-NumSeq is distributed in the hope that it will be useful, but
132       WITHOUT ANY WARRANTY; without even the implied warranty of
133       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
134       General Public License for more details.
135
136       You should have received a copy of the GNU General Public License along
137       with Math-NumSeq.  If not, see <http://www.gnu.org/licenses/>.
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141perl v5.30.0                      2019-08-05        Math::NumSeq::Polygonal(3)
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