 Pentagonal number

The $n$th pentagonal number is $P_n=n(3n-1)/2$ for a positive integer $n$, giving $1,5,12,22,\dots$. A positive integer $x$ is pentagonal exactly when $24x+1$ is a perfect square and $(1+\sqrt{24x+1})/6$ is an integer, by the defining quadratic equation.

A pentagonal number is a figurate number that represents a pentagon. The \( n \)-th pentagonal number can be calculated using the formula: \[ P(n) = \frac{n(3n - 1)}{2} \] where \( n \) is a positive integer. The sequence of pentagonal numbers begins with 1, 5, 12, 22, 35, and so on.