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On Divisibility of
Fibonomial Coefficients by $3$ Diego Marques
Departamento de Matemática
Universidade de Brasília
Brasília, DF
Brazil
mailto:diego@mat.unb.brdiego@mat.unb.br

Pavel Trojovský
Department of Mathematics
University of Hradec Králové
Faculty of Science
Rokitanského 62
Hradec Králové, 500 03
Czech Republic
mailto:pavel.trojovsky@uhk.czpavel.trojovsky@uhk.cz

in

Abstract:

Let $F_n$ be the $n$th Fibonacci number. For $1\le k\le m-1$ let

$${m\brack k}_F= \frac{F_m F_{m-1}\cdots F_{m-k+1}}{F_1\cdots F_k}$$ (1)

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 particular, among other things, we prove that $3 \mid {3^{a+1} \brack 3^a}_F$ , for all $a\geq 1$.