Definition:Hexagonal Number

Definition
Hexagonal numbers are those denumerating a collection of objects which can be arranged in the form of a regular hexagon. They are otherwise called hexagon numbers.

Or we can just say that a number is hexagonal.

They can be denoted $H_1, H_2, H_3, \ldots$, and they are formally defined as:
 * $\displaystyle H_n = \sum_{i \mathop = 1}^n \left({4 \left({i - 1}\right) + 1}\right) = 1 + 5 + \cdots + \left({4 \left({n-2}\right) + 1}\right) + \left({4 \left({n - 1}\right) + 1}\right)$

Recurrence Formula
It can be seen directly from the above that:
 * $H_n = \begin{cases}

0 & : n = 0 \\ H_{n-1} + 4 \left({n-1}\right) + 1 & : n > 0 \end{cases}$

Closed Form
From Closed Form for Polygonal Number‎s, we have (after some algebra):
 * $H_n = n \left({2 n - 1}\right)$

Also known as
Hexagonal numbers are also known as hexagonal numbers.

Or we can just say that a number is hexagonal.