Equivalence of Definitions of Hexagonal Number

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:
 * $\ds H_0 = \sum_{i \mathop = 1}^0 \paren {4 \paren {i - 1} + 1} = 0$

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:
 * $\ds H_0 = \sum_{i \mathop = 1}^0 \paren {4 \paren {i - 1} + 1}$

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 $H_0 = 0 = \map P {6, 0}$.

Then:

Thus $\map P {6, n}$ and $H_n$ are generated by the same recurrence relation.