"Binomiacci" Sum

Calculus Level 3

Let $\{a_0, a_1, a_2 \ldots, a_n\}$ be the set of integers that satisfy $\sum_{k=0}^{n} \binom{n}{k}a_k = F_n\ \ \forall n \in \{1,2,3,4,\ldots \},$ where $F_n$ is the $n^\text{th}$ Fibonacci number $(F_1=1, F_2=1, F_3=2, \ldots, F_{n+1} = F_n + F_{n-1}).$ Then we have $\lim_{n\rightarrow\infty} \frac{a_{n+1}}{a_n} = \frac{a+\sqrt{b}}{c},$ where $a$ , $b$ , and $c$ are integers with $a>0$ and $b$ square-free. What is $a+b+c?$