Definition:Generalized Pentagonal Number

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Definition

Definition 1

Recall the sequence of pentagonal numbers:

$0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, \ldots$

and the sequence of second pentagonal numbers:

$0, 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, \ldots$


The sequence of generalized pentagonal numbers consists of the elements of both of these sequences merged and arranged into ascending order.


Definition 2

The generalized pentagonal numbers are the integers obtained from the formula:

$GP_n = \begin{cases} \dfrac {m \left({3 m + 1}\right)} 2 & : n = 2 m \\ \dfrac {m \left({3 m - 1}\right)} 2 & : n = 2 m - 1 \end{cases}$

for $n = 0, 1, 2, \ldots$


Sequence

The sequence of generalized pentagonal numbers, for $n \in \Z_{\ge 0}$, begins:

$GP_n = 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, \ldots$

This sequence is A001318 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).