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,

                             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
                               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

Second Kind

Option 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.

Both Kinds

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), ...


Option 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 pairs option (a string) can be


Random Access

$value = $seq->ith($i)

Return the $i'th polygonal value, of the given pairs type.

$bool = $seq->pred($value)

Return true if $value is a polygonal number, of the given pairs type.

$i = $seq->value_to_i_estimate($value)

Return an estimate of the i corresponding to $value.


Math::NumSeq, Math::NumSeq::Cubes



Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2018, 2019, 2020, 2021 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 <>.