Sequence of Differences on Generalized Pentagonal Numbers

Theorem
Recall the generalised pentagonal numbers $GP_n$ for $n = 0, 1, 2, \ldots$

Consider the sequence defined as $\Delta_n = GP_{n + 1} - GP_n$:
 * $1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, \ldots$

Then:
 * The values of $\Delta_n$ for odd $n$ consist of the odd numbers
 * The values of $\Delta_n$ for even $n$ consist of the natural numbers.

Proof
Recall the definition of the generalised pentagonal numbers $GP_n$ for $n = 0, 1, 2, \ldots$
 * $GP_n = \begin{cases} \dfrac {m \paren {3 m + 1} } 2 & : n = 2 m \\

\dfrac {m \paren {3 m - 1} } 2 & : n = 2 m - 1 \end{cases}$ for $n = 0, 1, 2, \ldots$

Hence:

which defines the sequence of natural numbers.

Then:

which defines the sequence of odd numbers.