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<H1>Fibonacci numbers</H1>
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(definition)

<P>
<strong>Definition:</strong>
A sequence of numbers such that each number is the sum of the preceding two.   The first seven numbers are 1, 1, 2, 3, 5, 8, and 13. Formally, F(n) = F(n-1) + F(n-2), where F(1)=1 and F(2)=1.  F(n)  <img src="approx.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/approx.gif" border=0 height=6 width=13 alt="approximately equal to"> round( <img src="Phi.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/Phi.gif"    border=0 height=10 width=9  alt="Captial Phi"><sup>n</sup>/ <img src="sqrt.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/sqrt.gif"   border=0 height=15 width=12 alt="square root of"> 5), where  <img src="Phi.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/Phi.gif"    border=0 height=10 width=9  alt="Captial Phi">=(1+ <img src="sqrt.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/sqrt.gif"   border=0 height=15 width=12 alt="square root of"> 5)/2.
<P><strong>See also</strong>
<a href="kthordrfibon.html" tppabs="http://hissa.nist.gov/dads/HTML/kthordrfibon.html"><em>kth order Fibonacci numbers</em></a>, <a href="memoize.html" tppabs="http://hissa.nist.gov/dads/HTML/memoize.html"><em>memoization</em></a>.
<P><em>Note:
Fibonacci, or more correctly Leonardo of Pisa, discovered the series in 1202 when he was studying how fast rabbits could breed in ideal circumstances. <P> An exact formula for the n<sup>th</sup> Fibonacci number is F(n)= ( <img src="Phi.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/Phi.gif"    border=0 height=10 width=9  alt="Captial Phi"><sup>n</sup> - (- <img src="phi.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/phi.gif"    border=0 height=10 width=9  alt="phi">)<sup>n</sup>)/ <img src="sqrt.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/sqrt.gif"   border=0 height=15 width=12 alt="square root of"> 5, where  <img src="Phi.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/Phi.gif"    border=0 height=10 width=9  alt="Captial Phi">=(1+ <img src="sqrt.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/sqrt.gif"   border=0 height=15 width=12 alt="square root of"> 5)/2 and - <img src="phi.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/phi.gif"    border=0 height=10 width=9  alt="phi">=(1- <img src="sqrt.gif" tppabs="http://hissa.nist.gov/dads/HTML/Images/sqrt.gif"   border=0 height=15 width=12 alt="square root of"> 5)/2. <P> Computing Fibonacci numbers is an example in the notes for <a href="memoize.html" tppabs="http://hissa.nist.gov/dads/HTML/memoize.html"><em>memoization</em></a>.</em>
<P>Author: <a href="terms.html#authorPR" tppabs="http://hissa.nist.gov/dads/terms.html#authorPR">PR</a>
<H2>More information</H2>
A wonderful site on <A HREF="javascript:if(confirm('http://www.nist.gov/cgi-bin/exit_nist.cgi?url=http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/  \n\nThis file was not retrieved by Teleport Pro, because it is addressed on a domain or path outside the boundaries set for its Starting Address.  \n\nDo you want to open it from the server?'))window.location='http://www.nist.gov/cgi-bin/exit_nist.cgi?url=http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/'" tppabs="http://www.nist.gov/cgi-bin/exit_nist.cgi?url=http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/">Fibonacci Numbers and the Golden Section</A>.

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Entry modified Fri Jan  7 09:54:31 2000.<BR>
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