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The Fibonacci numbers Fn are given by the recurrence

The first few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... .

The generating function for the Fibonacci numbers is z/(1-z-z2), which can be proved easily from the recurrence (see GeneratingFunctions). Expanding the generating function using partial fractions gives the formula

\[F_n = \frac{1}{\sqrt{5}}\left(\phi^n - (1-\phi)^n\right),\]


\[\phi = \frac{1 + \sqrt{5}}{2}\]

is the golden_ratio.

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2014-06-17 11:58