Stuck Simplifying for a Fibonacci Series

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.

• Are you familiar with the method of generating functions? – Sean Roberson May 18 '18 at 17:06
• I don't understand the question. Suppose $g_n=F_n$, the usual Fibonacci sequence. What is your solution in this case? What is it you are even solving for? – lulu May 18 '18 at 17:30
• Voting to close the question as it is not clear what you are asking. If you can, please edit for clarity. Maybe work an example of what you are trying to solve for. – lulu May 18 '18 at 18:07