\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{Barbero}
\citation{Nemeth_binom}
\citation{Pan}
\citation{Spivey}
\citation{Bel2008}
\HyPL@Entry{0<</S/D>>}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}}
\newlabel{eq:trinomial_trans}{{1}{1}{Introduction}{section.1}{}}
\newlabel{eq:trinom}{{1}{1}{Introduction}{section.1}{}}
\citation{Sloane}
\citation{Nemeth_binom}
\citation{ANVI}
\citation{Andrews}
\citation{Bel2008}
\newlabel{eq:defsum}{{1}{2}{Introduction}{equation.1.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Construction of the trinomial transform triangle ${\cal  T}$}}{2}{figure.1}}
\newlabel{fig:construction}{{1}{2}{Construction of the trinomial transform triangle \T }{figure.1}{}}
\citation{Sloane}
\citation{Bel2008}
\@writefile{toc}{\contentsline {section}{\numberline {2}Trinomial triangle and its partial sum triangle}{3}{section.2}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Trinomial triangle (\href  {https://oeis.org/A027907}{\relax $\@@underline {\hbox {A027907}}\mathsurround \z@ $\relax })}}{3}{figure.2}}
\newlabel{fig:trinom}{{2}{3}{Trinomial triangle (\seqnum {A027907})}{figure.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Partial sum triangle of trinomial triangle}}{4}{figure.3}}
\newlabel{fig:trinom_subsum}{{3}{4}{Partial sum triangle of trinomial triangle}{figure.3}{}}
\newlabel{eq:n_n1}{{2}{4}{Trinomial triangle and its partial sum triangle}{equation.2.2}{}}
\newlabel{eq:nksum}{{3}{4}{Trinomial triangle and its partial sum triangle}{equation.2.2}{}}
\newlabel{th:sum_trinom_triangle}{{1}{4}{}{theorem.1}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Trinomial transform triangle}{5}{section.3}}
\newlabel{th:base}{{2}{5}{}{theorem.2}{}}
\newlabel{eq:base}{{4}{5}{}{equation.3.4}{}}
\newlabel{eq:sumrowdef}{{5}{6}{Trinomial transform triangle}{equation.3.5}{}}
\newlabel{eq:sumrowtri}{{4}{6}{}{theorem.4}{}}
\newlabel{th:sorsum}{{5}{6}{}{theorem.5}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Trinomial transform triangles generated by ternary homogeneous linear recurrent sequences}{7}{section.4}}
\newlabel{eq:def_ternary_seq}{{7}{7}{Trinomial transform triangles generated by ternary homogeneous linear recurrent sequences}{equation.4.7}{}}
\newlabel{eq:rec_abc}{{8}{8}{}{equation.4.8}{}}
\newlabel{th:rec_ABC}{{10}{8}{}{theorem.10}{}}
\newlabel{eq:rec_ABC}{{9}{8}{}{equation.4.9}{}}
\citation{Barbero1}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces A part of the growing of ${\cal  T}$}}{9}{figure.4}}
\newlabel{fig:steps_trinomial}{{4}{9}{A part of the growing of \T }{figure.4}{}}
\newlabel{th:trasform_of_ternary}{{11}{9}{Main corollary}{theorem.11}{}}
\citation{Barbero1}
\citation{sho}
\newlabel{th:column}{{13}{10}{}{theorem.13}{}}
\newlabel{eq:g3_abc}{{10}{10}{Trinomial transform triangles generated by ternary homogeneous linear recurrent sequences}{equation.4.10}{}}
\newlabel{eq:gk_abc}{{11}{10}{Trinomial transform triangles generated by ternary homogeneous linear recurrent sequences}{equation.4.11}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Sums and alternating sums of columns}{11}{subsection.4.1}}
\newlabel{eq:sn}{{12}{11}{}{equation.4.12}{}}
\newlabel{eq:altsumrow}{{4.1}{12}{Sums and alternating sums of columns}{equation.4.12}{}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Examples}{12}{section.5}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Fibonacci sequence}{13}{subsection.5.1}}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Fibonacci trinomial transform triangle}}{13}{table.1}}
\newlabel{tab:fib}{{1}{13}{Fibonacci trinomial transform triangle}{table.1}{}}
\newlabel{th:fib01}{{16}{13}{}{theorem.16}{}}
\newlabel{eq:fib_af}{{16}{13}{}{theorem.16}{}}
\newlabel{eq:fib_diag}{{17}{14}{}{theorem.17}{}}
\newlabel{eq:fib_nat}{{13}{14}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Tribonacci sequence}{14}{subsection.5.2}}
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Tribonacci trinomial transform triangle}}{15}{table.2}}
\newlabel{tab:tribo}{{2}{15}{Tribonacci trinomial transform triangle}{table.2}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}The constant sequence 1}{15}{subsection.5.3}}
\@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces Trinomial transform triangle generated by the constant sequence 1}}{16}{table.3}}
\newlabel{tab:const}{{3}{16}{Trinomial transform triangle generated by the constant sequence 1}{table.3}{}}
\newlabel{eq:sum_const}{{13}{16}{The constant sequence 1}{}{}}
\newlabel{eq:altsum_const}{{13}{16}{The constant sequence 1}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Natural numbers}{16}{subsection.5.4}}
\newlabel{eq:sum_natural}{{13}{16}{Natural numbers}{}{}}
\newlabel{eq:altsum_natural}{{13}{16}{Natural numbers}{}{}}
\bibcite{Andrews}{1}
\@writefile{lot}{\contentsline {table}{\numberline {4}{\ignorespaces Trinomial transform triangle generated by natural numbers}}{17}{table.4}}
\newlabel{tab:natur}{{4}{17}{Trinomial transform triangle generated by natural numbers}{table.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.5}Two other sequences}{17}{subsection.5.5}}
\bibcite{ANVI}{2}
\bibcite{Barbero1}{3}
\bibcite{Barbero}{4}
\bibcite{Bel2008}{5}
\bibcite{Nemeth_binom}{6}
\bibcite{Pan}{7}
\bibcite{Sloane}{8}
\bibcite{sho}{9}
\bibcite{Spivey}{10}