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\vskip 1cm{\LARGE\bf On Divisibility of \\
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Fibonomial Coefficients by $3$}
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\large
Diego Marques\\
Departamento de Matem\' atica\\
Universidade de Bras\' ilia\\
Bras\' ilia, DF \\
Brazil\\
\href{mailto:diego@mat.unb.br}{\tt diego@mat.unb.br} \\
\ \\
Pavel Trojovsk\' y\\
Department of Mathematics\\
University of Hradec Kr\'alov\'e\\
Faculty of Science \\
Rokitansk\'eho 62\\
Hradec Kr\'alov\'e, 500 03 \\
Czech Republic\\
\href{mailto:pavel.trojovsky@uhk.cz}{\tt pavel.trojovsky@uhk.cz}
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\begin{abstract}
Let $F_n$ be the $n$th Fibonacci number. For $1\le k\le
m-1$ let
\begin{equation}\label{def}
{m\brack k}_F= \frac{F_m F_{m-1}\cdots F_{m-k+1}}{F_1\cdots F_k}
\end{equation}
be the corresponding Fibonomial coefficient. In this paper, we present some divisibility properties of ${sn \brack n}_F$ by $3$, for some positive integers $n$ and $s$. In par\-ti\-cu\-lar, among other things, we prove that $3 \mid {3^{a+1} \brack 3^a}_F$ , for all $a\geq 1$.
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