\relax 
\@writefile{toc}{\contentsline {chapter}{\numberline {1}Background on Continued Fractions}{3}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {1.1}An Introduction to Continued Fractions}{3}}
\citation{Poorten1}
\citation{Poorten1}
\@writefile{toc}{\contentsline {section}{\numberline {1.2}Known Results}{4}}
\newlabel{corresp}{{1.2.1}{4}}
\newlabel{cor}{{1.1}{4}}
\newlabel{zero}{{1.2.2}{5}}
\newlabel{finitecf}{{1.2.3}{5}}
\citation{mw}
\citation{Poorten2}
\newlabel{pqneg}{{1.2.4}{6}}
\@writefile{toc}{\contentsline {section}{\numberline {1.3}Some Previous Techniques}{6}}
\citation{Poorten2}
\citation{Poorten1}
\citation{Poorten2}
\citation{Poorten2}
\newlabel{Uni}{{1.3.2}{7}}
\newlabel{delta}{{1.5}{8}}
\citation{Poorten2}
\citation{Poorten2}
\citation{Gosper}
\newlabel{beta2}{{1.6}{9}}
\@writefile{toc}{\contentsline {section}{\numberline {1.4}The Simple Model}{10}}
\citation{Oconnor}
\@writefile{toc}{\contentsline {chapter}{\numberline {2}Background on Fibonacci Numbers}{11}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {2.1}Brief History on Fibonacci Numbers}{11}}
\newlabel{diff}{{2.2}{11}}
\citation{Bicknell}
\@writefile{toc}{\contentsline {section}{\numberline {2.2}Basic Properties}{12}}
\newlabel{first}{{2.2.1}{12}}
\newlabel{second}{{2.2.2}{12}}
\newlabel{third}{{2.2.3}{12}}
\newlabel{negative}{{2.2.4}{12}}
\newlabel{fifth}{{2.2.5}{13}}
\@writefile{toc}{\contentsline {chapter}{\numberline {3}The {Continued Fraction }Expansion of Ratios of Fibonacci k-Neighbours}{14}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {3.1}Experimental Results}{14}}
\@writefile{toc}{\contentsline {section}{\numberline {3.2}First Efforts}{15}}
\@writefile{toc}{\contentsline {section}{\numberline {3.3}A Proof in Full Generality For Fibonacci Ratios}{17}}
\newlabel{star}{{3.1}{17}}
\newlabel{split}{{3.2}{18}}
\newlabel{fibodd}{{3.3}{18}}
\newlabel{fibeven}{{3.4}{18}}
\@writefile{toc}{\contentsline {section}{\numberline {3.4}Analysis}{19}}
\newlabel{kodd}{{3.5}{19}}
\newlabel{keven}{{3.6}{19}}
\@writefile{toc}{\contentsline {chapter}{\numberline {4}The General Difference Equation}{20}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {4.1}Basic Properties}{20}}
\newlabel{GDE}{{4.1}{20}}
\newlabel{secondg}{{4.1.1}{20}}
\citation{Drmota}
\newlabel{thirdg}{{4.1.2}{21}}
\newlabel{sub}{{4.2}{21}}
\newlabel{subb}{{4.3}{21}}
\newlabel{starr}{{4.4}{21}}
\newlabel{negativeg}{{4.1.3}{22}}
\newlabel{product}{{4.5}{22}}
\newlabel{negativeresult}{{4.7}{23}}
\newlabel{fifthg}{{4.1.4}{23}}
\newlabel{starrr}{{4.9}{24}}
\newlabel{sum}{{4.9}{24}}
\@writefile{toc}{\contentsline {chapter}{\numberline {5}More Closed Form Expressions}{25}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\newlabel{starg}{{5.2}{26}}
\newlabel{splitg}{{5.3}{26}}
\@writefile{toc}{\contentsline {section}{\numberline {5.1}Case 1: $l=1$}{26}}
\newlabel{diffl1}{{5.4}{26}}
\@writefile{toc}{\contentsline {section}{\numberline {5.2}Case 2: $l= -1$}{27}}
\@writefile{toc}{\contentsline {section}{\numberline {5.3}Case 3: $l$ divides $m$}{28}}
\newlabel{divisible}{{5.8}{28}}
\newlabel{dl1}{{5.9}{28}}
\@writefile{toc}{\contentsline {chapter}{\numberline {6}On Repeating Blocks of Length Greater Than 2}{31}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {6.1}Repeating Blocks of Length 3}{31}}
\newlabel{3block1}{{6.1}{32}}
\newlabel{3block2}{{6.2}{32}}
\newlabel{cond3}{{6.3}{32}}
\newlabel{form1}{{6.4}{32}}
\newlabel{form2}{{6.5}{32}}
\@writefile{toc}{\contentsline {chapter}{\numberline {7}For Those Who Come After}{36}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {chapter}{\numberline {A}Tables: Continued Fraction Expansions of Ratios of Fibonacci $\ \ \ \ \ \ $ k-Neighbours}{38}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {chapter}{\numberline {B}Tables: Length of the Repeating Blocks of the Powers of $\tau $, where $\tau $ has the form: $[\overline  {a_0,a_1,\mathinner {\ldotp \ldotp \ldotp },a_n}]$}{43}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {chapter}{\numberline {C}Tables: Length of the Repeating Blocks of the Powers of $\tau $, where $\tau $ has the form: $[a_0,\overline  {a_1,\mathinner {\ldotp \ldotp \ldotp },a_n}]$}{46}}
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\bibcite{Bicknell}{BH}
\bibcite{class}{Class Notes}
\bibcite{Drmota}{DR}
\bibcite{Gosper}{BG}
\bibcite{mw}{mathworld.com}
\bibcite{Oconnor}{OR}
\bibcite{Poorten1}{AvP1}
\bibcite{Poorten2}{AvP2}