\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand*\HyPL@Entry[1]{}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}}
\newlabel{e-original}{{1}{1}{Introduction}{equation.1.1}{}}
\citation{OEIS}
\citation{Keel}
\citation{OEIS}
\citation{Manin}
\citation{GLS}
\citation{DR}
\citation{Zeng}
\citation{CG}
\citation{GLS}
\citation{GLS}
\citation{GLS}
\newlabel{e-chi1}{{2}{2}{Introduction}{equation.1.2}{}}
\citation{GLS}
\newlabel{thm1}{{1}{3}{}{theorem.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2}First Proof of Theorem \ref  {thm1}}{3}{section.2}}
\newlabel{firstproof}{{2}{3}{First Proof of Theorem \ref {thm1}}{section.2}{}}
\newlabel{e-fun}{{5}{3}{First Proof of Theorem \ref {thm1}}{equation.2.5}{}}
\newlabel{e-fun2}{{6}{3}{First Proof of Theorem \ref {thm1}}{equation.2.6}{}}
\citation{W}
\newlabel{Sdef}{{7}{4}{First Proof of Theorem \ref {thm1}}{equation.2.7}{}}
\newlabel{Slem}{{1}{4}{}{lemma.1}{}}
\newlabel{e-SG}{{8}{4}{First Proof of Theorem \ref {thm1}}{equation.2.8}{}}
\newlabel{Maplesolution}{{9}{5}{}{equation.2.9}{}}
\newlabel{Wsub}{{2}{5}{}{equation.2.9}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Second Proof of Theorem \ref  {thm1}}{5}{section.3}}
\newlabel{secondproof}{{3}{5}{Second Proof of Theorem \ref {thm1}}{section.3}{}}
\newlabel{L-Phi}{{2}{5}{}{lemma.2}{}}
\newlabel{e-Phi}{{10}{5}{}{equation.3.10}{}}
\newlabel{e-Zcoeffs}{{11}{5}{}{equation.3.11}{}}
\citation{G}
\citation{X}
\newlabel{e-factor}{{12}{6}{}{equation.3.12}{}}
\newlabel{L-noeven}{{4}{6}{}{lemma.4}{}}
\newlabel{thm2}{{2}{7}{}{theorem.2}{}}
\newlabel{e-H}{{13}{7}{}{equation.3.13}{}}
\newlabel{e-J}{{14}{7}{Second Proof of Theorem \ref {thm1}}{equation.3.14}{}}
\citation{OEIS}
\citation{OEIS}
\newlabel{e-trees}{{15}{8}{Second Proof of Theorem \ref {thm1}}{equation.3.15}{}}
\newlabel{e-Hex}{{16}{8}{Second Proof of Theorem \ref {thm1}}{equation.3.16}{}}
\citation{EC2}
\@writefile{toc}{\contentsline {section}{\numberline {4}Third proof of Theorem \ref  {thm1}}{9}{section.4}}
\newlabel{thirdproof}{{4}{9}{Third proof of Theorem \ref {thm1}}{section.4}{}}
\newlabel{T-exp}{{3}{9}{}{theorem.3}{}}
\newlabel{e-Z}{{17}{9}{}{equation.4.17}{}}
\newlabel{e-explicit}{{18}{9}{}{equation.4.18}{}}
\newlabel{e-Lag1}{{19}{9}{Third proof of Theorem \ref {thm1}}{equation.4.19}{}}
\newlabel{e-Lag2}{{20}{9}{Third proof of Theorem \ref {thm1}}{equation.4.20}{}}
\bibcite{CG}{1}
\bibcite{W}{2}
\bibcite{DR}{3}
\bibcite{G}{4}
\bibcite{GLS}{5}
\bibcite{Keel}{6}
\bibcite{Manin}{7}
\bibcite{OEIS}{8}
\bibcite{EC2}{9}
\bibcite{X}{10}
\bibcite{Zeng}{11}