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What is the generating function $G(z)$ for the sequence of Fibonacci numbers?
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The general form is $G(x) = \frac{x}{1-x-x^2}$

and after solving this using partial fraction, we will get

$f_n=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^n- (\frac{1-\sqrt{5}}{2})^n)$