Equivalence of Definitions of Generalized Pentagonal Number

Proof
We just need to check the sequence $GP_n$ defined in Definition 2 contains both sequences of pentagonal numbers and second pentagonal numbers, and is in ascending order.

From Closed Form for Pentagonal Numbers:
 * $P_n = \dfrac {n \paren {3 n - 1}} 2$

which are the odd terms (and the zeroth term) of $GP_n$.

From definition of second pentagonal numbers:
 * $P'_n = \dfrac {n \paren {3 n + 1}} 2$

which are the even terms of $GP_n$.

Finally we need to show that $GP_n$ is increasing.

We have:
 * $GP_{2 m - 1} = \dfrac {m \paren {3 m - 1}} 2 < \dfrac {m \paren {3 m + 1}} 2 = GP_{2 m}$

and:
 * $GP_{2 m} = \dfrac {m \paren {3 m + 1}} 2 < \dfrac {\paren {m + 1} \paren {3 m + 2}} 2 = GP_{2 m + 1}$

Hence the result.