\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{EJ1957}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}}
\newlabel{f_jacobsthal}{{1}{1}{Introduction}{equation.1.1}{}}
\citation{LC1959}
\citation{LC1960}
\citation{RG1974}
\citation{HT2012}
\citation{HT2012}
\citation{EJ1957}
\citation{HT2012}
\newlabel{def_tver}{{1}{2}{}{theorem.1}{}}
\newlabel{eq_S_ab}{{2}{2}{Introduction}{equation.1.2}{}}
\citation{OE}
\newlabel{def:first}{{2}{3}{}{theorem.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Main results}{3}{section.2}}
\newlabel{prop}{{3}{3}{}{theorem.3}{}}
\newlabel{maintheorem_g}{{4}{4}{}{theorem.4}{}}
\newlabel{eqxy}{{3}{4}{Main results}{equation.2.3}{}}
\newlabel{proof-f3-1}{{4}{4}{Main results}{equation.2.4}{}}
\newlabel{proof-f3-2}{{5}{4}{Main results}{equation.2.5}{}}
\newlabel{proof-f3-3}{{6}{4}{Main results}{equation.2.6}{}}
\newlabel{proof-f4-1}{{7}{5}{Main results}{equation.2.7}{}}
\newlabel{proof-f4-2}{{8}{5}{Main results}{equation.2.8}{}}
\newlabel{proof-f4-3}{{9}{5}{Main results}{equation.2.9}{}}
\newlabel{proof-f4-4}{{10}{5}{Main results}{equation.2.10}{}}
\newlabel{proof-f4-5}{{11}{5}{Main results}{equation.2.11}{}}
\newlabel{proof-f4-6}{{12}{5}{Main results}{equation.2.12}{}}
\newlabel{proof-f4-7}{{13}{5}{Main results}{equation.2.13}{}}
\newlabel{eqx1x2x3-1}{{14}{5}{Main results}{equation.2.14}{}}
\newlabel{proof-1}{{15}{5}{Main results}{equation.2.15}{}}
\newlabel{proof-2}{{16}{5}{Main results}{equation.2.16}{}}
\newlabel{eq:sums_s_l}{{17}{6}{Main results}{equation.2.17}{}}
\newlabel{neweq1}{{18}{7}{Main results}{equation.2.18}{}}
\newlabel{eq:maxming1}{{19}{8}{Main results}{equation.2.19}{}}
\newlabel{eq:1+x^n}{{20}{8}{Main results}{equation.2.20}{}}
\newlabel{eq:-1^k-1nchoosek}{{21}{8}{Main results}{equation.2.21}{}}
\citation{EJ1957}
\citation{HT2012}
\newlabel{eq:maxming2}{{22}{9}{Main results}{equation.2.22}{}}
\newlabel{eq:maxming3}{{23}{9}{Main results}{equation.2.23}{}}
\newlabel{J_n,j_n}{{24}{9}{Main results}{equation.2.24}{}}
\newlabel{lem:faqm}{{7}{10}{}{theorem.7}{}}
\newlabel{(i)}{{7}{10}{}{theorem.7}{}}
\newlabel{maintheorem_f}{{8}{11}{}{theorem.8}{}}
\newlabel{eq:prooff-1}{{25}{11}{Main results}{equation.2.25}{}}
\newlabel{eq:f2-1}{{26}{11}{Main results}{equation.2.26}{}}
\newlabel{eq:f2-2}{{27}{11}{Main results}{equation.2.27}{}}
\newlabel{proof-fa1-a3-1}{{28}{11}{Main results}{equation.2.28}{}}
\newlabel{proof-fa1-a3-2}{{29}{11}{Main results}{equation.2.29}{}}
\newlabel{proof-fa1-a3-3}{{30}{11}{Main results}{equation.2.30}{}}
\newlabel{proof-fa1-a3-4}{{31}{12}{Main results}{equation.2.31}{}}
\newlabel{proof-fa1-a4-1}{{32}{12}{Main results}{equation.2.32}{}}
\newlabel{proof-fa1-a4-2}{{33}{12}{Main results}{equation.2.33}{}}
\newlabel{proof-fa1-a4-3}{{34}{12}{Main results}{equation.2.34}{}}
\newlabel{proof-fa1-a4-4}{{35}{12}{Main results}{equation.2.35}{}}
\newlabel{proof-fa1-a4-5}{{36}{12}{Main results}{equation.2.36}{}}
\newlabel{proof-fa1-a4-6}{{37}{12}{Main results}{equation.2.37}{}}
\newlabel{proof-fa1-a4-7}{{38}{12}{Main results}{equation.2.38}{}}
\newlabel{proof-fa1-a4-8}{{39}{12}{Main results}{equation.2.39}{}}
\newlabel{eq:proof-f-s1}{{40}{13}{Main results}{equation.2.40}{}}
\newlabel{eq:proof-f-n-1brcket}{{41}{13}{Main results}{equation.2.41}{}}
\newlabel{eq:maxmin-f}{{42}{14}{Main results}{equation.2.42}{}}
\newlabel{neqtheorem9}{{9}{14}{}{theorem.9}{}}
\newlabel{abcd}{{43}{14}{}{equation.2.43}{}}
\citation{EJ1957}
\citation{HT2012}
\newlabel{0<S_a,b;m<m/2}{{44}{15}{Main results}{equation.2.44}{}}
\newlabel{proof-S3-1}{{45}{15}{Main results}{equation.2.45}{}}
\newlabel{proof-S3-2}{{46}{15}{Main results}{equation.2.46}{}}
\newlabel{proof-S3-3}{{47}{15}{Main results}{equation.2.47}{}}
\newlabel{proof-S3}{{48}{15}{Main results}{equation.2.48}{}}
\newlabel{proof-S4-1}{{49}{16}{Main results}{equation.2.49}{}}
\newlabel{proof-S4-2}{{50}{16}{Main results}{equation.2.50}{}}
\newlabel{proof-S4-3}{{51}{16}{Main results}{equation.2.51}{}}
\newlabel{proof-S4-4}{{52}{16}{Main results}{equation.2.52}{}}
\newlabel{proof-S4-5}{{53}{16}{Main results}{equation.2.53}{}}
\newlabel{proof-S4-6}{{54}{16}{Main results}{equation.2.54}{}}
\newlabel{proof-S4-7}{{55}{16}{Main results}{equation.2.55}{}}
\newlabel{proof-S4}{{56}{16}{Main results}{equation.2.56}{}}
\newlabel{proof-S-s1}{{57}{16}{Main results}{equation.2.57}{}}
\newlabel{proof-S-s1-2}{{58}{17}{Main results}{equation.2.58}{}}
\newlabel{-sum<S<sum}{{59}{18}{Main results}{equation.2.59}{}}
\bibcite{LC1959}{1}
\bibcite{LC1960}{2}
\bibcite{RG1974}{3}
\bibcite{EJ1957}{4}
\bibcite{OE}{5}
\bibcite{HT2012}{6}
\@writefile{toc}{\contentsline {section}{\numberline {3}Acknowledgments}{19}{section.3}}