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fibonacci [2019/03/31 14:49] (current)
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 +====== 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
  • Last modified: 2019/03/31 14:49
  • (external edit)