# Differences

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 — fibonacci [2019/03/31 14:49] (current) Line 1: Line 1: + ====== Fibonacci Numbers ====== + + **Fibonacci numbers** are defined by the following recurrance: + + $F_0\ =\ 0,$\\ + $F_1\ =\ 1,$\\ + $F_i\ =\ F_{ i-1 } +\ F_{ i-2 }\ for\ i\geq 2.$ + + Each fibonacci number is the sum of the two previous ones, yielding the sequence + + $0,​1,​1,​2,​3,​5,​8,​13,​21,​34,​55,​ ...$ + + Fibonacci numbers are related to the **golden ratio** $\phi$ and to its + conjugate $\hat{\phi}$ which are the two roots of the equation: + + $x^2\ =\ x\ +\ 1$ + + and are given by the following: + + $\phi = \frac{1+\sqrt{ 5 }}{2}$\\ + $\hat{ \phi } = \frac{1-\sqrt{ 5 }}{2}$ + + Where we have: + + $F_i = \frac{\phi ^i-\hat{\phi}^i}{\sqrt{5}}$ + + Fibonacci numbers grow exponentially. +
• fibonacci.txt