How to tell if a Fibonacci number has an even or odd index Given only $F_n$, that is the $n$th term of the Fibonacci sequence, how can you tell if $n \equiv 1 \mod 2$ or $n \equiv 0 \mod 2$?
I know you can use the Pisano period, however if  $n \equiv 1$ or $n \equiv 2$ $\mod \pi(k)$, it can never be found, where $k$ is in $\pi(k)$ (The Pisano period).
Also there is the fact that if $\sqrt{5F_n^2+ 4}$ is an integer then $n 
\equiv 0 \mod 2$, but is there a faster way?
Lastly, because $F_1 = F_2 = 1$, that would have to be an exception for whatever rule/formula that would apply.

Assuming that $F_n\geq 2$, you can check the parity of $n$ depending on the sign of the difference between $\frac{1+\sqrt{5}}{2}F_n$ and the closest integer. If negative, then $n$ is even, if positive, then $n$ is odd.
