Definition:Hexagonal Number

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Definition

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


They can be denoted:

$H_1, H_2, H_3, \ldots$


Definition 1

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


Definition 2

$\displaystyle H_n = \sum_{i \mathop = 1}^n \left({4 \left({i - 1}\right) + 1}\right) = 1 + 5 + \cdots + \left({4 \left({n-2}\right) + 1}\right) + \left({4 \left({n - 1}\right) + 1}\right)$


Definition 3

$\forall n \in \N: H_n = P \left({6, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({6, n - 1}\right) + 4 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$

where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.


Examples of Hexagonal Numbers

The first few hexagonal numbers are as follows:


HexagonNumbers.png


Sequence of Hexagonal Numbers

The sequence of hexagonal numbers, for $n \in \Z_{\ge 0}$, begins:

$0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, \ldots$


Also known as

Hexagonal numbers are also known as hexagon numbers.

Or we can just say that a number is hexagonal.


Some sources denote the $n$th hexagonal number as $\map H n$ in preference to $H_n$.


Also see

  • Results about hexagonal numbers can be found here.


Sources