Definition:Generalized Pentagonal Number/Definition 2

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Definition

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:

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


Also see

  • Results about generalized pentagonal numbers can be found here.


Sources