Pentagonal number

The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex.

The nth pentagonal number is the sum of n integers starting from n (i.e. from n to 2n-1).

This division of centered hexagonal arrays gives generalized pentagonal numbers as trapezoidal arrays, which may be interpreted as Ferrers diagrams for their partition.

For generalized pentagonal numbers, it is sufficient to just check if 24x + 1 is a perfect square.

For non-generalized pentagonal numbers, in addition to the perfect square test, it is also required to check if The mathematical properties of pentagonal numbers ensure that these tests are sufficient for proving or disproving the pentagonality of a number.

A visual representation of the first six pentagonal numbers
Proof without words that the n th pentagonal number can be decomposed into three triangular numbers and the number n .