Equivalence of Definitions of Octagonal Number

Definition 1 implies Definition 2
Let $O_n$ be an octagonal number by definition 1.

Let $n = 0$.

By definition:
 * $O_0 = 0$

By vacuous summation:
 * $\ds O_0 = \sum_{i \mathop = 1}^0 \paren {6 i - 5} = 0$

By definition of summation:

and so:

Thus $O_n$ is an octagonal number by definition 2.

Definition 2 implies Definition 1
Let $O_n$ be an octagonal number by definition 2.

Then:

Then:
 * $\ds O_0 = \sum_{i \mathop = 1}^0 \paren {6 i - 5}$

is a vacuous summation and so:


 * $O_0 = 0$

Thus $O_n$ is an octagonal number by definition 1.

Definition 1 equivalent to Definition 3
We have by definition that $O_n = 0 = \map P {8, n}$.

Then:

Thus $\map P {8, n}$ and $O_n$ are generated by the same recurrence relation.