How do you determine if a number is a even Fibonacci number or not? Rather than computing out the whole Fibonacci sequence and check if $n$ is even and in there, is there a more straightforward way to compute if $n$ is a even Fibonacci number?

The $n^\text {th} $ Fibonacci number is given by $$ F_n=\frac {\phi^n-(-\phi)^{-n}}{\sqrt {5}}, $$ where $\phi $ is the golden ratio. If your number is $x $, then you could try $$\frac {\cosh^{-1}(\sqrt {5}x/2)}{\ln\phi}.$$ If this is an even integer, then $x $ is a Fibonacci number. Also if $$\frac {\sinh^{-1}(\sqrt {5}x/2)}{\ln\phi}$$ is odd.
