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Find: $$ \sum_{n=0}^\infty \sum_{k=0}^n \frac{F_{2k}F_{n-k}}{10^n} $$ where $F_n$ is $n$-th Fibonacci number.

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Have you used generating functions before? If so, you just need to find the ordinary generating functions for $F_n$ and $F_{2n}$. Then multiply them and plug in $x = 1/10$. –  SpamIAm Mar 31 at 19:25
    
This has a basic answer at this MSE link. –  Marko Riedel Aug 5 at 22:18

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