Pentagonal number
= Pentagonal number
{title2=$P_n$}
The $n$th <pentagonal number> is $P_n=n(3n-1)/2$ for a <positive integer> $n$, giving $1,5,12,22,\ldots$. 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>.