Definition:Generalized Pentagonal Number/Definition 1

Definition
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.

Also see

 * Equivalence of Definitions of Generalized Pentagonal Number