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Antti H S Laaksonen 2017-02-25 16:51:29 +02:00
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@ -778,7 +778,7 @@ n! & = & n \cdot (n-1)! \\
\index{Fibonacci number}
The \key{Fibonacci numbers} arise in many situations.
The \key{Fibonacci numbers}\footnote{Fibonacci (c. 1175--1250) was an Italian mathematician.} arise in many situations.
They can be defined recursively as follows:
\[
\begin{array}{lcl}
@ -790,7 +790,8 @@ f(n) & = & f(n-1)+f(n-2) \\
The first Fibonacci numbers are
\[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots\]
There is also a closed-form formula
for calculating Fibonacci numbers:
for calculating Fibonacci numbers\footnote{This formula is sometimes called
\index{Binet's formula} \key{Binet's formula}.}:
\[f(n)=\frac{(1 + \sqrt{5})^n - (1-\sqrt{5})^n}{2^n \sqrt{5}}.\]
\subsubsection{Logarithms}