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. In general the k-gonals for k>=3 are
P(i) = (k-2)/2 * i*(i+1) - (k-3)*i
pairs => 'second' gives the polygonals of the second kind, which are the same formula negating the i.
S(i) = (k-2)/2 * i*(i-1) + (k-3)*i
The result is positive values, bigger than the plain polygonals. 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 is given just once at i=0,
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 if k is odd but i is even. If both k and i odd then it's an 0.5.
See "FUNCTIONS" in Math::NumSeq for the behaviour common to all path classes.
$seq = Math::NumSeq::Polygonal->new (key=>value,...)
Create and return a new sequence object.
$value = $seq->ith($i)
$i ** 2.
$bool = $seq->pred($value)
Return true if
$valueis a perfect square.
Copyright 2010, 2011 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/>.