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.

It remains to be demonstrated that $8 F_n + 1$ is square :
 * $n \in \set{\pm 1, 0, 2, 4, 8, 10}$

So, let $8 F_n + 1$ be square.

Then:
 * $n \equiv \begin{cases} \pm 1 \pmod {2^5 \times 5} & : n \text { odd} \\ 0, 2, 4, 8, 10 \pmod {2^5 \times 5^2 \times 11} & : n \text { even} \end{cases}$