Square of Odd Number as Difference between Triangular Numbers

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

Then:
 * $\exists a, b \in \Z_{\ge 0}: \paren {2 n + 1}^2 = T_a - T_b$

where:
 * $T_a$ and $T_b$ are triangular numbers
 * $T_a$ and $T_b$ are coprime.

That is, the square of every odd number is the difference between two coprime triangular numbers.

Proof
Let $a = 3b + 1$

The square of every odd number can be made through this method.

Both triangular numbers are coprime.