\begin{definition}[Definition:Generalized Pentagonal Number/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 $n = 0, 1, 2, \ldots$
\end{definition}