Codex @codex  0

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 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.