You wish they were multiplied

$\displaystyle f(n+2)=\frac { f(n+1)+f(n) }{ \gcd(f(n),f(n+1)) }$

A function $f : \mathbb{N} \to \mathbb{N}$ satisfies the above equation for all $n \in \mathbb{N}$ and it has an upper bound. Find the sum of all possible values of $f(2016)$ .

Clarification : $\mathbb{N}$ denotes the set of positive integers $\{1, 2, 3, \ldots\}$ .