\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]{}
\citation{hds}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}}
\newlabel{sec1}{{1}{1}{Introduction}{section.1}{}}
\newlabel{eq1}{{1}{1}{Introduction}{equation.1.1}{}}
\citation{hhg}
\citation{sna}
\citation{gk}
\citation{sna}
\citation{cct}
\citation{ktv}
\citation{mcl1}
\citation{hhg}
\newlabel{eq2}{{2}{2}{Introduction}{equation.1.2}{}}
\newlabel{eq3}{{3}{2}{Introduction}{equation.1.3}{}}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces First 200 terms of Hofstadter's $Q(n)$}}{3}{table.1}}
\newlabel{tab11}{{1}{3}{First 200 terms of Hofstadter's $Q(n)$}{table.1}{}}
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces First 200 terms of $W(n)$}}{3}{table.2}}
\newlabel{tab12}{{2}{3}{First 200 terms of $W(n)$}{table.2}{}}
\@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces First 200 terms of $V(n)$}}{3}{table.3}}
\newlabel{tab13}{{3}{3}{First 200 terms of $V(n)$}{table.3}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Graph of first 200 values of Hofstadter's $Q(n)$}}{4}{figure.1}}
\newlabel{fig11}{{1}{4}{Graph of first 200 values of Hofstadter's $Q(n)$}{figure.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Graph of first 200 values of $W(n)$}}{4}{figure.2}}
\newlabel{fig12}{{2}{4}{Graph of first 200 values of $W(n)$}{figure.2}{}}
\citation{hhg}
\citation{cct}
\citation{hts2}
\citation{tsm}
\citation{pks1}
\@writefile{toc}{\contentsline {section}{\numberline {2}Monotonicity}{5}{section.2}}
\newlabel{sec2}{{2}{5}{Monotonicity}{section.2}{}}
\newlabel{thm21}{{1}{5}{}{theorem.1}{}}
\newlabel{eq21}{{4}{5}{}{equation.2.4}{}}
\newlabel{eq22}{{5}{5}{}{equation.2.5}{}}
\newlabel{eq23}{{6}{5}{Monotonicity}{equation.2.6}{}}
\newlabel{eq24}{{7}{6}{Monotonicity}{equation.2.7}{}}
\newlabel{eq25}{{8}{6}{Monotonicity}{equation.2.8}{}}
\newlabel{eq26}{{9}{6}{Monotonicity}{equation.2.9}{}}
\newlabel{eq27}{{10}{6}{Monotonicity}{equation.2.10}{}}
\newlabel{eq28}{{11}{6}{Monotonicity}{equation.2.11}{}}
\newlabel{eq29}{{12}{7}{Monotonicity}{equation.2.12}{}}
\newlabel{eq210}{{13}{7}{Monotonicity}{equation.2.13}{}}
\newlabel{eq211}{{14}{7}{Monotonicity}{equation.2.14}{}}
\newlabel{eq212}{{15}{7}{Monotonicity}{equation.2.15}{}}
\newlabel{eq213}{{16}{7}{Monotonicity}{equation.2.16}{}}
\newlabel{eq214}{{17}{7}{Monotonicity}{equation.2.17}{}}
\newlabel{eq215}{{18}{7}{Monotonicity}{equation.2.18}{}}
\newlabel{eq216}{{19}{7}{Monotonicity}{equation.2.19}{}}
\newlabel{eq217}{{20}{8}{Monotonicity}{equation.2.20}{}}
\newlabel{eq218}{{21}{8}{Monotonicity}{equation.2.21}{}}
\newlabel{col22}{{2}{9}{}{theorem.2}{}}
\newlabel{col24}{{3}{9}{}{theorem.3}{}}
\newlabel{col25}{{4}{9}{}{theorem.4}{}}
\newlabel{pps26}{{5}{9}{}{theorem.5}{}}
\citation{ktv}
\citation{tsm}
\citation{gk}
\newlabel{col27}{{6}{10}{}{theorem.6}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Frequency Sequence Properties}{10}{section.3}}
\newlabel{sec3}{{3}{10}{Frequency Sequence Properties}{section.3}{}}
\@writefile{lot}{\contentsline {table}{\numberline {4}{\ignorespaces First 200 values of frequency sequence $F(a)$ of $V(n)$}}{11}{table.4}}
\newlabel{tab31}{{4}{11}{First 200 values of frequency sequence $F(a)$ of $V(n)$}{table.4}{}}
\newlabel{lem31}{{7}{11}{}{theorem.7}{}}
\citation{hhg}
\newlabel{lem32}{{8}{12}{}{theorem.8}{}}
\newlabel{lem33}{{9}{12}{}{theorem.9}{}}
\newlabel{lem34}{{10}{12}{}{theorem.10}{}}
\newlabel{eq31}{{23}{12}{Frequency Sequence Properties}{equation.3.23}{}}
\newlabel{lem35}{{11}{12}{}{theorem.11}{}}
\newlabel{eq32}{{24}{13}{Frequency Sequence Properties}{equation.3.24}{}}
\newlabel{lem36}{{12}{13}{}{theorem.12}{}}
\newlabel{lem37}{{13}{14}{}{theorem.13}{}}
\newlabel{lem38}{{14}{14}{}{theorem.14}{}}
\newlabel{lem39}{{15}{15}{}{theorem.15}{}}
\newlabel{lem310}{{16}{15}{}{theorem.16}{}}
\newlabel{lem311}{{17}{15}{}{theorem.17}{}}
\newlabel{lem312}{{18}{15}{}{theorem.18}{}}
\newlabel{lem313}{{19}{16}{}{theorem.19}{}}
\@writefile{lot}{\contentsline {table}{\numberline {5}{\ignorespaces Frequencies of $2a$ and $2a + 1$ in terms of the frequencies of $a$ and some of its neighbors.}}{16}{table.5}}
\newlabel{tab32}{{5}{16}{Frequencies of $2a$ and $2a + 1$ in terms of the frequencies of $a$ and some of its neighbors}{table.5}{}}
\citation{hhg}
\@writefile{lot}{\contentsline {table}{\numberline {6}{\ignorespaces Values of the frequency sequence at the start and endpoints of the intervals [$2^k$,$2^{k + 1} - 1$].}}{17}{table.6}}
\newlabel{tab33}{{6}{17}{Values of the frequency sequence at the start and endpoints of the intervals [$2^k$,$2^{k + 1} - 1$]}{table.6}{}}
\citation{hhg}
\@writefile{lot}{\contentsline {table}{\numberline {7}{\ignorespaces Gutman's rules for generating the frequency sequence of $V(n)$}}{18}{table.7}}
\newlabel{tab34}{{7}{18}{Gutman's rules for generating the frequency sequence of $V(n)$}{table.7}{}}
\citation{cct}
\citation{ktv}
\citation{mcl1}
\citation{pks1}
\citation{pks2}
\citation{tsm}
\citation{pks1}
\citation{pks2}
\citation{gkp}
\citation{dbs}
\@writefile{toc}{\contentsline {section}{\numberline {4}V-sequence Generational Structure}{19}{section.4}}
\newlabel{sec4}{{4}{19}{V-sequence Generational Structure}{section.4}{}}
\newlabel{eq41}{{25}{19}{V-sequence Generational Structure}{equation.4.25}{}}
\newlabel{pps41}{{20}{19}{}{theorem.20}{}}
\@writefile{lot}{\contentsline {table}{\numberline {8}{\ignorespaces The first 100 values of $M(n)$}}{20}{table.8}}
\newlabel{tab41}{{8}{20}{The first 100 values of $M(n)$}{table.8}{}}
\@writefile{lot}{\contentsline {table}{\numberline {9}{\ignorespaces Maternal generation structure of $V(n)$ and the frequencies of the $V$-values of the start and end points of first 18 generations.}}{21}{table.9}}
\newlabel{tab42}{{9}{21}{Maternal generation structure of $V(n)$ and the frequencies of the $V$-values of the start and end points of first 18 generations}{table.9}{}}
\newlabel{lem42}{{21}{21}{}{theorem.21}{}}
\newlabel{pps43}{{22}{21}{}{theorem.22}{}}
\newlabel{pps44}{{23}{22}{}{theorem.23}{}}
\newlabel{pps45}{{24}{23}{}{theorem.24}{}}
\newlabel{col46}{{25}{24}{}{theorem.25}{}}
\citation{cct}
\citation{hts1}
\citation{hts2}
\citation{ktv}
\citation{tsm}
\@writefile{toc}{\contentsline {section}{\numberline {5}Alternative Initial Conditions}{26}{section.5}}
\newlabel{sec5}{{5}{26}{Alternative Initial Conditions}{section.5}{}}
\@writefile{lot}{\contentsline {table}{\numberline {10}{\ignorespaces Recursion \textup  {\hbox {\mathsurround \z@ \normalfont  (\ignorespaces \ref  {eq3}\unskip \@@italiccorr )}} with alternative initial conditions that result in a well defined sequence}}{26}{table.10}}
\newlabel{tab51}{{10}{26}{Recursion \eqref {eq3} with alternative initial conditions that result in a well defined sequence}{table.10}{}}
\newlabel{pps51}{{26}{26}{}{theorem.26}{}}
\newlabel{eq51}{{26}{27}{Alternative Initial Conditions}{equation.5.26}{}}
\newlabel{eq52}{{27}{27}{Alternative Initial Conditions}{equation.5.27}{}}
\newlabel{eq53}{{28}{27}{Alternative Initial Conditions}{equation.5.28}{}}
\newlabel{eq54}{{29}{27}{Alternative Initial Conditions}{equation.5.29}{}}
\bibcite{cct}{1}
\bibcite{dbs}{2}
\bibcite{gkp}{3}
\bibcite{gk}{4}
\bibcite{hts1}{5}
\@writefile{lot}{\contentsline {table}{\numberline {11}{\ignorespaces Examples of recursion \textup  {\hbox {\mathsurround \z@ \normalfont  (\ignorespaces \ref  {eq3}\unskip \@@italiccorr )}} with alternative initial conditions that result in sequences that die after a very long life}}{28}{table.11}}
\newlabel{tab52}{{11}{28}{Examples of recursion \eqref {eq3} with alternative initial conditions that result in sequences that die after a very long life}{table.11}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6}Concluding Remarks}{28}{section.6}}
\newlabel{sec6}{{6}{28}{Concluding Remarks}{section.6}{}}
\newlabel{eq61}{{30}{28}{Concluding Remarks}{equation.6.30}{}}
\newlabel{ack}{{6}{28}{Acknowledgements}{section*.1}{}}
\bibcite{hts2}{6}
\bibcite{hds}{7}
\bibcite{hhg}{8}
\bibcite{ktv}{9}
\bibcite{mcl1}{10}
\bibcite{pks1}{11}
\bibcite{pks2}{12}
\bibcite{sna}{13}
\bibcite{tsm}{14}