수열의 귀납적 정의

Is the following conjecture true or false?

Let $f_1, f_2, \ldots, f_m, \ldots$ be the sequence of the Fibonacci numbers. For each natural number $n$, the numbers $f_nf_{n+3}$, $2f_{n+1}f_{n+2}$, and $\left( f_{n+1}^2 + f_{n+2}^2 \right)$ form a Pythagorean triple.