# "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 $$a+b+c?$$

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