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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.

For integers $s\ge 3$ and $n\ge 1$, the $n$th $s$-gonal number is $P_{s,n}=((s-2)n^2-(s-4)n)/2$. Its successive differences form an arithmetic progression. The cases $s=3,4,5,6$ give triangular numbers, square numbers, pentagonal numbers, and hexagonal numbers.