On Divisibility of Fibonomial Coefficients by 3

Abstract:

Let

be the

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

, for some positive integers

and

. 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.