\begin{definition}[Definition:Generalized Pentagonal Number/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.
\end{definition}