"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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