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Let (Fn)n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as [ m k ] F = Fm−k+1 · · ·Fm−1Fm F1 · · ·Fk , and [ m k ] F = 0, for k > m. In this paper, we shall prove that if p is a prime number such that p ≡ −2 or 2 (mod 5), then p | [ p pa ]
New results about certain sums Sn(k) of products of the Lucas numbers are derived. These sums are related to the generating function of the k-th powers of the Fibonacci numbers. The sums for Sn(k) are expressed by the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special inverse formula.
Fibonacci sequence (or sequence of the Fibonacci numbers) 〈Fn〉∞n=0 is the sequence of positive integers satisfying the recurrence relation Fn+2 = Fn+1 +Fn with the initial conditions F0 = 0 and F1 = 1. Lucas sequence is a sequence 〈Ln〉∞n=0 of positive integers satisfying the same recurrence as the Fibonacci numbers but with the initial conditions L0 = 2 and… (More)
The aim of this paper is to give new results about factorizations of the Fibonacci numbers Fn and the Lucas numbers Ln. These numbers are defined by the second order recurrence relation an+2 = an+1+an with the initial terms F0 = 0, F1 = 1 and L0 = 2, L1 = 1, respectively. Proofs of theorems are done with the help of connections between determinants of… (More)
The aim of the paper is to find some new determinants connected with Fibonacci numbers. We generalize the result provided in Strang’s book because we derive that two sequences of similar tridiagonal matrices are connected with Fibonacci numbers. AMS subject classification: Primary 15A15, 11B39; Secondary 11B37, 11B83.