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
- $H_n = \begin{cases} 0 & : n = 0 \\ H_{n - 1} + 4 \paren {n - 1} + 1 & : n > 0 \end{cases}$
Definition 2
- $\ds H_n = \sum_{i \mathop = 1}^n \paren {4 \paren {i - 1} + 1} = 1 + 5 + \cdots + \paren {4 \paren {n - 2} + 1} + \paren {4 \paren {n - 1} + 1}$
Definition 3
- $\forall n \in \N: H_n = \map P {6, n} = \begin{cases} 0 & : n = 0 \\ \map P {6, n - 1} + 4 \paren {n - 1} + 1 & : n > 0 \end{cases}$
where $\map P {k, n}$ denotes the $k$-gonal numbers.
Examples of Hexagonal Numbers
The first few hexagonal numbers are as follows:
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
- Closed Form for Hexagonal Numbers: $H_n = n \paren {2 n - 1}$
- Results about hexagonal numbers can be found here.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $45$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $45$