TITLE: Is there a graphical proof that $9n+1$ is triangular if $n$ is?
QUESTION [5 upvotes]: It is straightforward to prove that if $n$ is a triangular number then $9n+1$ must be.
Is there a systematic decomposition of a triangle made of $9n+1$ dots into nine triangles of $n$ dots plus one left over to illustrate this fact visually?

REPLY [7 votes]: Another one, based on the fact that centered hexagonal numbers can be expressed as $6T_n+1$: