Square of Odd Multiple of 3 is Difference between Triangular Numbers

Theorem
Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $T_n$ denote the $n$th triangular number.

Let $m = 2 n + 1$ be an odd integer

Then:


 * $\paren {3 m}^2 = T_{9 n + 4} - T_{3 n + 1}$