Journal of Integer Sequences, Vol. 15 (2012), Article 12.6.4

On Divisibility of Fibonomial Coefficients by 3

Diego Marques
Departamento de Matemática
Universidade de Brasília
Brasília, DF

Pavel Trojovský
Department of Mathematics
University of Hradec Králové
Faculty of Science
Rokitanského 62
Hradec Králové, 500 03
Czech Republic


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

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

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(Concerned with sequences A000045 A144712.)

Received March 1 2012; revised version received June 11 2012. Published in Journal of Integer Sequences, June 19 2012.

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