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-When talking about recurisve sequences there is a poster child, the fibbonacci sequence. $F_{n} = F_{n-1} + F_{n-2}$, but did you know there is a closed, non-recursive, $O(1)$, formula for the fibbonacci numbers $F_n = \frac{\varphi^n-\psi^n}{\sqrt 5}$. Where $\varphi = bla$ and $\psi = bla$. This blog post will explore several ways to prove this identity, in order to not only verify the formula, but see how you could have discovered it, as well as uncovering a fun connection to a $O(n)$ algorithm and linear algebra.
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