<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%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 $i$,
                      $1 \leq i &lt; m$ and $k_m \geq 2$.}
                     {\HSpace} 
         &amp;
         \FibProperty{Definitions:}%
                     {$\begin{array}{l@{\hspace{.1em}}c@{\hspace{.2em}}l}
                          %Line 1
                          F_0 &amp;=&amp; F_1 = 1\\
                          %Line 2
                          F_i &amp;=&amp; F_{i-1} + F_{i-2}\\
                          %Line 3
                          F_{-i} &amp;=&amp; (-1)^{i-1} \\[\VSpace]
                          %Line 4
                          \rule{0pt}{3.5ex plus .1ex minus 1ex}% To increase space bewtween array's lines
                          \phi &amp;=&amp; \frac{1+\sqrt{5}}{2}\text{,} \quad \hat\phi = \frac{1-\sqrt{5}}{2} = 1-\phi\\
                          %Line 5 
                          \rule{0pt}{3.5ex plus .1ex minus 1ex}% To increase space bewtween array's lines
                          F_i    &amp;=&amp; \frac{1}{\sqrt{5}} \left(\phi^i - \hat{\phi}^i\right)\\
                       \end{array}$}
                     {\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} 
         &amp;
         \FibProperty{Cassini's identity for $i &gt; 0$:}
                     {$F_{i+1} F_{i-1} - F^2_i = (-1)^i$}
                     {\HSpace} 
         \\ %Line 3
         \FibProperty{Additive rule:}%
                     {$\begin{array}{lcl}%
                          %Line 1
                          F_{n+k} &amp;=&amp; F_k F_{n+1} + F_{k-1} F_n \\
                          %Line 2
                          F_{2n} &amp;=&amp; F_n F_{n+1} + F_{n-1} F_n
                       \end{array}$}
                     {\HSpace}  
         &amp;
         \rule{0pt}{3ex plus 1ex minus .5ex}%Add a little bit space between the 
                                            %lines of the array
         \FibProperty{Calculation by matrices:}
                     {$\begin{pmatrix}
                        F_{n-2} &amp;F_{n-1} \\
                        F_{n-1} &amp;F_n \\
                     \end{pmatrix}
                     =
                     \begin{pmatrix}
                        0 &amp;1 \\
                        1 &amp;1 \\
                     \end{pmatrix}^n$}
                     {\HSpace} 
   \end{tabular}
   }%
}
%This is the title of this part
\newcommand\TTenFibTitle{The Fibonacci numbers}
</pre></body></html>