Definition:Pentagonal Number

Definition
Pentagonal numbers are those denumerating a collection of objects which can be arranged in the form of a regular pentagon.

They are otherwise called pentagon numbers.

Or we can just say that a number is pentagonal.

They can be denoted $P_1, P_2, P_3, \ldots$, and they are formally defined as:
 * $\displaystyle P_n = \sum_{i \mathop = 1}^n \left({3 \left({i - 1}\right) + 1}\right) = 1 + 4 + \cdots + \left({3 \left({n-2}\right) + 1}\right) + \left({3 \left({n-1}\right) + 1}\right)$.

Thus $P_0 = 0$ and $P_1 = 1$.


 * [[File:2ndPentagonalNumber.png]]

The second pentagonal number: $P_2 = 1 + 4 = 5$.


 * [[File:3rdPentagonalNumber.png]]

The third pentagonal number: $P_3 = 1 + 4 + 7 = 12$.

Recurrence Formula
It can be seen directly from the above that:
 * $P_n = \begin{cases}

0 & : n = 0 \\ P_{n-1} + 3 \left({n-1}\right) + 1 & : n > 0 \end{cases}$

Closed Form
From Closed Form for Polygonal Number‎s, we have:
 * $P_n = \dfrac {n \left({2 + 3 \left({n-1}\right)}\right)} 2 = \dfrac {n \left({3 n - 1}\right)} 2$