\contentsline {chapter}{\numberline {1}Background on Continued Fractions}{3}
\contentsline {section}{\numberline {1.1}An Introduction to Continued Fractions}{3}
\contentsline {section}{\numberline {1.2}Known Results}{4}
\contentsline {section}{\numberline {1.3}Some Previous Techniques}{6}
\contentsline {section}{\numberline {1.4}The Simple Model}{10}
\contentsline {chapter}{\numberline {2}Background on Fibonacci Numbers}{11}
\contentsline {section}{\numberline {2.1}Brief History on Fibonacci Numbers}{11}
\contentsline {section}{\numberline {2.2}Basic Properties}{12}
\contentsline {chapter}{\numberline {3}The {Continued Fraction }Expansion of Ratios of Fibonacci k-Neighbours}{14}
\contentsline {section}{\numberline {3.1}Experimental Results}{14}
\contentsline {section}{\numberline {3.2}First Efforts}{15}
\contentsline {section}{\numberline {3.3}A Proof in Full Generality For Fibonacci Ratios}{17}
\contentsline {section}{\numberline {3.4}Analysis}{19}
\contentsline {chapter}{\numberline {4}The General Difference Equation}{20}
\contentsline {section}{\numberline {4.1}Basic Properties}{20}
\contentsline {chapter}{\numberline {5}More Closed Form Expressions}{25}
\contentsline {section}{\numberline {5.1}Case 1: $l=1$}{26}
\contentsline {section}{\numberline {5.2}Case 2: $l= -1$}{27}
\contentsline {section}{\numberline {5.3}Case 3: $l$ divides $m$}{28}
\contentsline {chapter}{\numberline {6}On Repeating Blocks of Length Greater Than 2}{31}
\contentsline {section}{\numberline {6.1}Repeating Blocks of Length 3}{31}
\contentsline {chapter}{\numberline {7}For Those Who Come After}{36}
\contentsline {chapter}{\numberline {A}Tables: Continued Fraction Expansions of Ratios of Fibonacci $\ \ \ \ \ \ $ k-Neighbours}{38}
\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}
\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}