Sequence of Differences on Generalized Pentagonal Numbers

Theorem
Take the sequence of pentagonal numbers and the sequence of second pentagonal numbers and arrange them all into ascending order:
 * $P_n = 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, \ldots$

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

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
Consider the sequence $P_n$:
 * $P_n = 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, \ldots$

where $P_0 = 0$ is that first element of the sequence.

It is seen that:
 * for even $n$, $P_n$ is the $\dfrac n 2$th second pentagonal number
 * for odd $n$, $P_n$ is the $\dfrac {n + 1} 2$th pentagonal number.

So from the definition of the second pentagonal numbers:
 * $P_{2 n} = \dfrac {n \left({3 n + 1}\right)} 2$

and from Closed Form for Pentagonal Numbers:
 * $P_{2 n - 1} = \dfrac {n \left({3 n - 1}\right)} 2$

Hence:

which defines the sequence of natural numbers.

Then:

which defines the sequence of odd numbers.