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:

$P_1, P_2, P_3, \ldots$


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:

$P' \left({n}\right) = \dfrac {n \left({3 n + 1}\right)} 2$

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