Equivalence of Definitions of Generalized Pentagonal Number

Theorem

The following definitions of the concept of Generalized Pentagonal Number are equivalent:

$0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, \ldots$

$0, 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, \ldots$

The sequence of generalized pentagonal numbers consists of the elements of both of these sequences merged and arranged into ascending order.

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 $n = 0, 1, 2, \ldots$

It is sufficient to verify that the sequence $GP_n$ defined in Definition 2 contains both sequences of pentagonal numbers and second pentagonal numbers, and is in ascending order.

$P_n = \dfrac {n \paren {3 n - 1} } 2$

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

$P'_n = \dfrac {n \paren {3 n + 1} } 2$

which are the even terms of $GP_n$.

Finally it is shown 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.

$\blacksquare$