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\begin{document}

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\theoremstyle{plain}
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\begin{center}
\vskip 1cm{\LARGE\bf 
On Functions Expressible as Words  \\
\vskip .1in
on a Pair of Beatty Sequences
}
\vskip 1cm
\large
Christian  Ballot \\
D\'epartement de Math\'ematiques et Informatique\\
Universit\'e de Caen-Normandie \\
France \\
\href{mailto:christian.ballot@unicaen.fr}{\tt christian.ballot@unicaen.fr} \\
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\def\cI{{\mathcal I}}
\def\cR{{\mathcal R}}


\def\a{{\alpha}}
\def\b{{\beta}}
\def\g{{\gamma}}
\def\d{{\delta}}
\def\l{{\lambda}}
\def\o{{\omega}}
\def\e{{\epsilon}}
\def\ep{{\varepsilon}}
\def\s{{\sigma}}
\def\t{{\tau}}
\def\v{{\nu}}
\def\th{{\theta}}


\def \K{{\bbbk}}
\def\E{{\mathbf E}}
\def\G{{\mathcal G}}
\def\O{{\mathcal O}}
\def \R{{\bbbr}}
\def\({\left(}
\def\){\right)}

\def\lf{\lfloor}
\def\rf{\rfloor}
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\begin{abstract}  
Let $a(n)=\lfloor n\a\rfloor$ and $b(n)=\lfloor n\a^2\rfloor$, where
$\a=\frac{1+\sqrt{5}}2$. Then a theorem of Kimberling states that each
function $f$, composed of several $a$'s and $b$'s, can be expressed in
the form $c_1a+c_2b-c_3,$ where $c_1$ and $c_2$ are consecutive
Fibonacci numbers determined by the numbers of $a$'s and of $b$'s
composing $f$ and $c_3$ is a nonnegative constant. We provide
generalizations of this theorem to two infinite families of
complementary pairs of Beatty sequences.  The particular case involving
`Narayana' numbers is examined in depth. The details reveal that $x_n=
\lf\a^3\lf\a^3\lf\cdots\lf\a^3\rf\cdots\rf\rf\rf$, with $n$ nested
pairs of $\lf\;\rf$, is a 7th-order linear recurrence, where $\a$ is the
dominant zero of $x^3-x^2-1$.
\end{abstract}


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