# Algorithm Wiki

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# Fibonacci

The Fibonacci number or sequence is:

$0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\;$

Each number is the sum of the previous two numbers. In mathematical terms:

$F_n = F_{n-1} + F_{n-2}$

where

$F_1 = 1,\; F_2 = 1$

## Recursive

The recursive algorithm to calculate a fibonacci number is very simple and is a direct translation from the mathematical definition. Unfortunately it is very slow. It uses $O(\phi^n)$ time, where $\phi=\frac{\sqrt{5}+1}{2}$ (the golden ratio).

WARNING: Javascript math uses floating point for integers. After fib(78) the Javascript implementation of these algorithms are not exact.

## Dynamic programming

Dynamic programming is the other obvious solution. The algorithm calculates each fibonacci number in sequence. This algorithm uses $O(n)$ time.

## Fast doubling

\begin{aligned} F_{2n} & = F_{n}(2*F_{n+1}-F_{n}) \\ F_{2n+1} & = F_{n+1}^2+F{n}^2 \end{aligned}