Equivalence of Definitions of Octagonal Number

Theorem
The following definitions of an octagonal number are equivalent:

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:
 * $\displaystyle O_0 = \sum_{i \mathop = 1}^0 \left({6 \left({i - 1}\right) + 1}\right) = 0$

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:
 * $\displaystyle O_0 = \sum_{i \mathop = 1}^0 \left({6 \left({i - 1}\right) + 1}\right)$

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 $O_n = 0 = P \left({8, n}\right)$.

Then:

Thus $P \left({8, n}\right)$ and $O_n$ are generated by the same recurrence relation.