<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% m211oh78.tex (prev.: oh74.*) - Fibonacci Numbers
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\begin{center}\tcr{\rmH Fibonacci Numbers} \end{center} %\vspace{0pt}
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\begin{description}  

\item[\tcg{History:}]
-named after \tcr{Fibonacci} (= filio Bonacci) or \tcr{Leonardo da Pisa} 
\tcg{(1170?-1240?)}, who wrote the book \tcr{Liber Abbaci} (= book of computation) in \tcg{1202}.  

\item[\tcg{Definition:}] -via the ($2^{nd}$ order) 
\tcr{recursion relation}: %diff. eq'n):
$$
\tcb{a_{\scr n+2} = a_{\scr n+1} + a_n,\quad a_{\scr0} = 0, a_{\scr1} = 1.}
$$
\tcr{$\Rightarrow$} the first few numbers are \mc{0, 1, 1, 2, 3, 5, 8, 13, 21, \ldots}

\item[\tcg{Original Problem}] \tcr{(Fibonacci, 1202):} How many pairs of rabbits can be produced from a single pair in a year if each pair begets a new pair each month, which from the $2^{nd}$ month on becomes productive, and if death does not occur?

\item[\tcg{Notes:}] \tcg{1)} \tcb{Fibonacci numbers} have fascinated many people throughout the ages (particularly non-mathe\-ma\-ticians). There is a whole journal devoted to them (\tcr{Fibonacci Quarterly}).

\tcg{2)} \tcb{Fibonacci numbers} are closely related to phenomena involving the \tcr{golden ratio/section}.

\tcg{3)} \tcb{Fibonacci numbers} have \tcg{botanical applications} in the phenomenon called \tcr{phyllotaxis} (= leaf arrangement).   
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