"Binomiacci" Sum

Calculus Level 3

Let {a0,a1,a2,an}\{a_0, a_1, a_2 \ldots, a_n\} be the set of integers that satisfy k=0n(nk)ak=Fn  n{1,2,3,4,},\sum_{k=0}^{n} \binom{n}{k}a_k = F_n\ \ \forall n \in \{1,2,3,4,\ldots \}, where FnF_n is the nthn^\text{th} Fibonacci number (F1=1,F2=1,F3=2,,Fn+1=Fn+Fn1).(F_1=1, F_2=1, F_3=2, \ldots, F_{n+1} = F_n + F_{n-1}). Then we have limnan+1an=a+bc,\lim_{n\rightarrow\infty} \frac{a_{n+1}}{a_n} = \frac{a+\sqrt{b}}{c}, where aa, bb, and cc are integers with a>0a>0 and bb square-free. What is a+b+c?a+b+c?

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