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

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

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

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

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

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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.


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

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Visualization

Proof Without Words

Triangular Numbers

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