Equivalence of Definitions of Hexagonal Number

Theorem
The following definitions of a hexagonal number are equivalent:

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

Let $n = 0$.

By definition:
 * $H_0 = 0$

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

By definition of summation:

and so:

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

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

Then:

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

is a vacuous summation and so:


 * $H_0 = 0$

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

Definition 1 equivalent to Definition 3
We have by definition that $H_n = 0 = P \left({6, n}\right)$.

Then:

Thus $P \left({6, n}\right)$ and $H_n$ are generated by the same recurrence relation.