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%This command provides the Fibonacci definitions located in the third
%column of the third horizontal part of page 10
%
%The command has one parameter
% 1) The width of the text
\newcommand\TTenFibonacci[1]{%
%
%This macro will typeset a property of the Fibonacci number
%and the associated formulae.
%
%The macro has three parameters
% 1) The text of the property
% 2) The associated formula
% 3) The width to use to write the text.
\def\FibProperty##1##2##3{%
\parbox[t]{##3}{\noindent \textit{##1}\\##2}%
}%
\parbox[t]{#1}{%
\deflength{\HSpace}{.5#1}%
\begin{tabular}{wl{\HSpace}|wl{\HSpace}}%
\TTenFibonacciFontSize
\deflength{\HSpace}{\HSpace-2\tabcolsep}%
%Line 1
\FibProperty{The Fibonacci number system:}
{Every integer $n$ has a unique representation
\[
n = F_{k_1} + F_{k_2} + \cdots + F_{k_m}
\]
where $k_i \geq k_{i+1} + 2$ for all $i$,
$1 \leq i < m$ and $k_m \geq 2$.}
{\HSpace}
&
\FibProperty{Definitions:}%
{$\begin{array}{l@{\hspace{.1em}}c@{\hspace{.2em}}l}
%Line 1
F_0 &=& F_1 = 1\\
%Line 2
F_i &=& F_{i-1} + F_{i-2}\\
%Line 3
F_{-i} &=& (-1)^{i-1} \\[\VSpace]
%Line 4
\rule{0pt}{3.5ex plus .1ex minus 1ex}% To increase space bewtween array's lines
\phi &=& \frac{1+\sqrt{5}}{2}\text{,} \quad \hat\phi = \frac{1-\sqrt{5}}{2} = 1-\phi\\
%Line 5
\rule{0pt}{3.5ex plus .1ex minus 1ex}% To increase space bewtween array's lines
F_i &=& \frac{1}{\sqrt{5}} \left(\phi^i - \hat{\phi}^i\right)\\
\end{array}$}
{\HSpace}
\\ %Line 2
\rule{0pt}{3ex plus 1ex minus .5ex}%Add a little bit space between the
%lines of the array
\FibProperty{The first Fibonacci numbers:}
{$1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
89, \ldots$}
{\HSpace}
&
\FibProperty{Cassini's identity for $i > 0$:}
{$F_{i+1} F_{i-1} - F^2_i = (-1)^i$}
{\HSpace}
\\ %Line 3
\FibProperty{Additive rule:}%
{$\begin{array}{lcl}%
%Line 1
F_{n+k} &=& F_k F_{n+1} + F_{k-1} F_n \\
%Line 2
F_{2n} &=& F_n F_{n+1} + F_{n-1} F_n
\end{array}$}
{\HSpace}
&
\rule{0pt}{3ex plus 1ex minus .5ex}%Add a little bit space between the
%lines of the array
\FibProperty{Calculation by matrices:}
{$\begin{pmatrix}
F_{n-2} &F_{n-1} \\
F_{n-1} &F_n \\
\end{pmatrix}
=
\begin{pmatrix}
0 &1 \\
1 &1 \\
\end{pmatrix}^n$}
{\HSpace}
\end{tabular}
}%
}
%This is the title of this part
\newcommand\TTenFibTitle{The Fibonacci numbers}
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