Definition:Second Pentagonal Number
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Definition
The second pentagonal numbers are those denumerating a collection of objects which can be arranged in the form of a regular pentagon.
They can be denoted:
- $\map {P'} 1, \map {P'} 2, \map {P'} 3, \ldots$
Definition 1
The second pentagonal numbers are the integers obtained by applying the Closed Form for Pentagonal Numbers to negative $n$:
- $\map {P'} n = \dfrac {-n \paren {-3 n - 1} } 2$
Definition 2
The second pentagonal numbers are the integers obtained from formula:
- $\map {P'} n = \dfrac {n \paren {3 n + 1} } 2$
for $n = 0, 1, 2, \ldots$
Sequence
The sequence of second pentagonal numbers, for $n \in \Z_{\ge 0}$, begins:
- $0, 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, \ldots$
Also see
- Results about second pentagonal numbers can be found here.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $22$