Equivalence of Definitions of Heptagonal Number

Theorem
The following definitions of a heptagonal number are equivalent:

Definition 1 implies Definition 2
Let $H_n$ be a heptagonal number by definition 1.

Let $n = 0$.

By definition:
 * $H_0 = 0$

By vacuous summation:
 * $\displaystyle H_0 = \sum_{i \mathop = 1}^0 \left({5 i - 4}\right) = 0$

By definition of summation:

and so:

Thus $H_n$ is a heptagonal number by definition 2.

Definition 2 implies Definition 1
Let $H_n$ be a heptagonal number by definition 2.

Then:

Then:
 * $\displaystyle H_0 = \sum_{i \mathop = 1}^0 \left({5 i - 4}\right)$

is a vacuous summation and so:


 * $H_0 = 0$

Thus $H_n$ is a heptagonal number by definition 1.

Definition 1 equivalent to Definition 3
We have by definition that $H_0 = 0 = P \left({7, 0}\right)$.

Then:

Thus $P \left({7, n}\right)$ and $H_n$ are generated by the same recurrence relation.