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Math::NumSeq::Polygonal -- polygonal numbers, triangular, square, pentagonal, etc
use Math::NumSeq::Polygonal; my $seq = Math::NumSeq::Polygonal->new (polygonal => 7); my ($i, $value) = $seq->next;
The sequence of polygonal numbers. The 3-gonals are the triangular numbers i*(i+1)/2, the 4-gonals are squares i*i, the 5-gonals are pentagonals (3i-1)*i/2, etc.
In general the k-gonals for k>=3 are
P(i) = (k-2)/2 * i*(i+1) - (k-3)*i
The values are how many points are in a triangle, square, pentagon, hexagon, etc of side i. For example the triangular numbers,
d c c d b b c b c d a a b a b c a b c d i=1 i=2 i=3 i=4 value=1 value=3 value=6 value=10
Or the squares,
d d d d c c c c c c d b b b b c b b c d a a b a b c a b c d i=1 i=2 i=3 i=4 value=1 value=4 value=9 value=16
Or pentagons (which should be a pentagonal grid, so skewing a bit here),
d d d c d c d c c d c c d b c b c c b c d b b b b c b b c d a a b a b c a b c d i=1 i=2 i=3 i=4 value=1 value=5 value=12 value=22
The letters "a", "b" "c" show the extra added onto the previous figure to grow its points. Each side except two are extended. In general the k-gonals increment by k-2 sides of i points, plus 1 at the end of the last side, so
P(i+1) = P(i) + (k-2)*i + 1
pairs => 'second' gives the polygonals of the second kind, which are the same formula but with a negative i.
S(i) = P(-i) = (k-2)/2 * i*(i-1) + (k-3)*i
The result is still positive values, bigger than the plain P(i). For example the pentagonals are 0,1,5,12,22,etc and the second pentagonals are 0,2,7,15,26,etc.
pairs => 'both' gives the firsts and seconds interleaved. P(0) and S(0) are both 0 and that value is given just once at i=0, so
0, P(1), S(1), P(2), S(2), P(3), S(3), ...
pairs => 'average' is the average of the first and second, which ends up being simply a multiple of the perfect squares,
A(i) = (P(i)+S(i))/2 = (k-2)/2 * i*i
This is an integer if k is even, or k odd and i is even. If k and i both odd then it's an 0.5 fraction.
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
$seq = Math::NumSeq::Polygonal->new ()
$seq = Math::NumSeq::Polygonal->new (pairs => $str)
Create and return a new sequence object. The default is the polygonals of the "first" kind, or the
pairsoption (a string) can be
"first" "second" "both" "average"
$value = $seq->ith($i)
$i'th polygonal value, of the given
$bool = $seq->pred($value)
Return true if
$valueis a polygonal number, of the given
$i = $seq->value_to_i_estimate($value)
Return an estimate of the i corresponding to
Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2018, 2019 Kevin Ryde
Math-NumSeq 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-NumSeq 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-NumSeq. If not, see <http://www.gnu.org/licenses/>.