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There are many nice ways of showing that $f_0^2+f_1^2+\cdots+f_n^2=f_{n+1}f_n$. I was wondering if there is a way of showing this using the generating function $F(x)=\frac{1}{1-x-x^2}=\sum_{i\geq0}f_ix^i$. In other words, is there any operation (perhaps the Hadamard product) that can be applied to $F(x)$ that would yield the identity above?

What about other identities that involve sums and squares, like $f_1f_2+\cdots +f_nf_{n+1}=f_{n+1}^2$ for $n$ odd?

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Apply the technique described in this blog post to $F(x) F(y)$, then to $x F(x) F(y)$. The key observation here is that one can compute Hadamard products of rational functions using the residue theorem.

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