I am attempting to solve for $n$ in the equation:
$g_n=g_1F_{n-1}+g_2F_n$
where $F_n$ is the $n$th Fibonacci number. I know that $g_0$ and $g_1$ will be positive integers such that $0 < g_1 \leq g_2$. I am given $g_n$ and $g_{n-1}$ to begin with, and I have the following identities to work with:
$g_n=g_{n-1}+g_{n-2}$
$F_n=\frac{\phi^n-\bar{\phi}^n}{\sqrt{5}}$
$\phi+\bar{\phi}=1$
$\phi\bar{\phi}=-1$
My current solution is very messy and, importantly, incomplete. I began by renaming a couple variables: $g_n=a$, $g_{n-1}=b$, and $g_{n-2}=c$. Which allows me to do the following:
$a=F_{n-1}g_1+F_ng_2=b+c$
and I used that to infer $b=F_ng_2$ and $c=F_{n-1}g_1$, but I do not have proof for that. I tried going from here to get rid of the terms $g_1$ and $g_2$, but this gets very messy very quickly once you begin to unpack everything. I really need some help figuring out where to go from here. Any advice at all would be greatly appreciated. I need to avoid the recursive definition of this function at all costs.