<s>[INST] Is there a Fibonacci identity? Let $F(n)$ be the n-th Fibonacci number. Is there any identity to simplify the following summation.
$$\sum_{i=1}^{n-1}F(i)^2F(n-i)^2 $$
 [/INST] $$F(i)=\frac{a^i-b^i}{a-b},\ a=\frac{1+\sqrt{5}}{2},\ b=\frac{1-\sqrt{5}}{2},\\ F^2(i)=\frac{a^{2i}+b^{2i}-2(-1)^i}{5}\implies S(x)=\sum_{i\ge 1}F^{2i}x^i=\frac{\sum_{i\ge 1}a^{2i}x^i+\sum_{i\ge 1}b^{2i}x^i-2\sum_{i\ge 1}(-x)^i}{5}=\frac{1}{5}\left(\frac{1}{1-a^2x}+\frac{1}{1-b^2x}-\frac{2}{1+x}\right)=\frac{1}{5}\left(\frac{x^2+5x}{1-2x-2x^2+x^3}\right)$$ Then the given expression will be the coefficient of $x^{n}$ in the expansion of $S^2(x)$
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