Definition:Pentagonal Number
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Definition
Pentagonal numbers are those denumerating a collection of objects which can be arranged in the form of a regular pentagon.
They can be denoted:
- $P_1, P_2, P_3, \ldots$
Definition 1
- $P_n = \begin{cases} 0 & : n = 0 \\ P_{n - 1} + 3 n - 2 & : n > 0 \end{cases}$
Definition 2
- $\ds P_n = \sum_{i \mathop = 1}^n \paren {3 i - 2} = 1 + 4 + \cdots + \paren {3 \paren {n - 1} - 2} + \paren {3 n - 2}$
Definition 3
- $\forall n \in \N: P_n = \map P {5, n} = \begin {cases} 0 & : n = 0 \\ \map P {5, n - 1} + 3 \paren {n - 1} + 1 & : n > 0 \end {cases}$
where $\map P {k, n}$ denotes the $k$-gonal numbers.
Examples of Pentagonal Numbers
The first few pentagonal numbers are as follows:
Sequence of Pentagonal Numbers
The sequence of pentagonal numbers, for $n \in \Z_{\ge 0}$, begins:
- $0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, \ldots$
Also known as
Pentagonal numbers are also known as pentagon numbers.
Or we can just say that a number is pentagonal.
Also see
- Closed Form for Pentagonal Numbers: $P_n = \dfrac {n \paren {3 n - 1} } 2$
- Results about pentagonal numbers can be found here.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $22$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): figurate numbers
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.13$: Fermat ($\text {1601}$ – $\text {1665}$)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $22$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): pentagonal number
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): pentagonal number