Is there a graphical proof that $9n+1$ is triangular if $n$ is?

Question

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?

Answer 1

Visual proof. ${}{}{}{}{}{}{}{}{}{}{}$

Of course it doesn't depend on the base triangle, here is the same with $T_4$:

With regard to the pattern mentionned by Teresa Lisbon in a comment above, here is how $25T_n+3$ shows up:

The pattern may be repeated with a number of "holes" that is itself a triangle number, for instance:

In light red, the repeated pattern.

This pattern yields the general result: $8T_pT_q+T_p+T_q=T_n$ with $n=2pq+p+q$, and of course $T_n=\frac12n(n+1)$. It's not difficult to prove algebraically.

Answer 2

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