Definition:Second Pentagonal Number

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

• Equivalence of Definitions of Second Pentagonal Number

• Definition:Pentagonal Number
• Definition:Generalized Pentagonal Number

• Results about second pentagonal numbers can be found here.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $22$