| Version 14 |
Version 13 |
| The \emph{Fibonacci sequence}, discovered by Leonardo Pisano Fibonacci, begins |
The \emph{Fibonacci sequence}, discovered by Leonardo Pisano Fibonacci, begins |
|
|
| $$ 0, 1, 1, 2, 3, 5 ,8 , 13, 21, 34, 55, 89, 144, 233, 377, \ldots $$ |
$$ 0, 1, 1, 2, 3, 5 ,8 , 13, 21, 34, 55, 89, 144, 233, 377, \ldots $$ |
|
|
| (Sequence \PMlinkexternal{A000045}{http://www.research.att.com/projects/OEIS?Anum=A000045} in \cite{OEIS}). |
(Sequence \PMlinkexternal{A000045}{http://www.research.att.com/projects/OEIS?Anum=A000045} in \cite{OEIS}). |
| The $n$th Fibonacci number is generated by adding the previous two. Thus, the Fibonacci sequence has the recurrence relation |
The $n$th Fibonacci number is generated by adding the previous two. Thus, the Fibonacci sequence has the recurrence relation |
|
|
| $$ f_n = f_{n-1} + f_{n-2} $$ |
$$ f_n = f_{n-1} + f_{n-2} $$ |
|
|
| with $f_0=0$ and $f_1 = 1$. This recurrence relation can be solved into the closed form |
with $f_0=0$ and $f_1 = 1$. This recurrence relation can be solved into the closed form |
|
|
| $$ f(n) = \frac{1}{\sqrt{5}} \left( \phi^n - \phi'^{\;n} \right) $$ |
$$ f(n) = \frac{1}{\sqrt{5}} \left( \phi^n - \phi'^{\;n} \right) $$ |
| called the Binet formula. |
called the Binet formula. |
|
|
| Where $\phi$ is the golden ratio (also see this entry for an explanation of $\phi'$.) Note that |
Where $\phi$ is the golden ratio (also see this entry for an explanation of $\phi'$.) Note that |
|
|
| $$ \lim_{n\rightarrow \infty} \frac{f_{n+1}}{f_n} = \phi. $$ |
$$ \lim_{n\rightarrow \infty} \frac{f_{n+1}}{f_n} = \phi. $$ |
|
|
| \begin{thebibliography}{15} |
\begin{thebibliography}{15} |
| \bibitem{OEIS} |
\bibitem{OEIS} |
| N. J. A. Sloane, (2004), The On-Line Encyclopedia of Integer Sequences, \PMlinkexternal{http://www.research.att.com/~njas/sequences/}{http://www.research.att.com/~njas/sequences/}. |
N. J. A. Sloane, (2004), The On-Line Encyclopedia of Integer Sequences, \PMlinkexternal{http://www.research.att.com/~njas/sequences/}{http://www.research.att.com/~njas/sequences/}. |
| \end{thebibliography} |
\end{thebibliography} |
