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The $n$th hexagonal number is $H_n=n(2n-1)$ for a positive integer $n$. Since $H_n=T_{2n-1}$, every hexagonal number is also a triangular number.

For integers $s\ge 3$ and $n\ge 1$, the $n$th $s$-gonal number isThese figurate numbers begin at one, and their successive increments form an arithmetic progression:Summing these increments proves the formula. For $s=3,4,5,6$, it gives triangular numbers, square numbers, pentagonal numbers and hexagonal numbers. For example, the first three pentagonal numbers are $1,5,12$, whose successive increments are four and seven. Increasing the number of polygon sides changes the step between increments while keeping this common quadratic structure.