Is there a closed form or series representation for this linear recurrence?

Question

Define the linear recurrence $a_n=F_na_{n-1}+a_{n-2}$ with $a_0=1$, $a_1=2$, and $F_n$ being the Fibonacci series ($F_0=F_1=1$). Is there a possible closed form for $a_n$, or even a series representation? I've looked at several resources, but many of them just deal with recurrences with constant coefficients.