Triangular Fibonacci Numbers

Theorem
The only Fibonacci numbers which are also triangular are:
 * $0, 1, 3, 21, 55$

Proof
It remains to be shown that these are the only ones.

Let $F_n$ be the $n$th Fibonacci number.

From Odd Square is Eight Triangles Plus One, $F_n$ is triangular $8 F_n + 1$ is square.