The Fibonacci sequence {Fn} is defined by the recurrence relation Fn = Fn−1+ Fn−2, for n ≥ 2 with F0 = 0 and F1 = 1. The Lucas sequence {Ln} , considered as a companion to Fibonacci sequence, is defined recursively by Ln = Ln−1 + Ln−2, for n ≥ 2 with L0 = 2 and L1 = 1. It is well known that F−n = (−1)Fn and L−n = (−1)Ln, for every n ∈ Z. For more detailed information see [9],[10]. This paper pr...

In this study, we define hyperbolic-type k-Fibonacci numbers and then give the relationships between k-step Fibonacci numbers. addition, study sequence modulo m periods of Hperbolic-type sequences for any k which are related m. Furthermore, extend to groups. Finally, obtain 2-Fibonacci in dihedral group D2m, (m ? 2) with respect generating pairs (x,y) (y, x).

We show that the p-adic valuation of the sequence of Fibonacci numbers is a p-regular sequence for every prime p. For p 6= 2, 5, we determine that the rank of this sequence is α(p) + 1, where α(m) is the restricted period length of the Fibonacci sequence modulo m.

We reduce the Fibonacci sequence mod m for a natural number m, and denote it by F (mod m). We are going to introduce the properties of the period and distribution of F (mod m). That is, how frequently each residue is expected to appear within a single period. These are well known themes of the research of the Fibonacci sequence, and many remarkable facts have been discovered. After that we are ...

An integer sequence (xn)n≥0 is said to be Fibonacci-like if it satisfies the binary recurrence relation xn = xn−1 + xn−2, n ≥ 2. We construct a new type of Fibonacci-like sequence of composite numbers. 1 The problem and previous results In this paper we consider Fibonacci-like sequences, that is, sequences (xn) ∞ n=0 satisfying the binary recurrence relation xn = xn−1 + xn−2, n ≥ 2. (1)

In this paper we obtain formulas for certain sums of products involving multinomial coefficients and Fibonacci numbers. The sums studied here may be regarded as generalizations of the binomial transform of the sequence comprising the even-numbered terms of the Fibonacci sequence. The general formulas, involving both Fibonacci and Lucas numbers, give rise to infinite sequences that are parameter...

The sequence obtained to solve this problem—the celebrated Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, ...—appears in a large number of natural phenomena (see [2], [6]) and has natural applications in computer science (see [1]). Here we reformulate the rabbit problem to recover two generalizations of the Fibonacci sequence presented elsewhere (see [7], [8]). Then, using a fixed-point technique...

The Fibonacci cube Γn is a subgraph of n-dimensional hypercube induced by the vertices without two consecutive ones. Klavžar and Žigert [Fibonacci cubes are the resonance graphs of fibonaccenes, Fibonacci Quart. 43 (2005) 269–276] proved that Fibonacci cubes are precisely the Z -transformation graphs (or resonance graphs) of zigzag hexagonal chains. In this paper, we characterize plane bipartit...

Let Γn and Λn be the n-dimensional Fibonacci cube and Lucas cube, respectively. The domination number γ of Fibonacci cubes and Lucas cubes is studied. In particular it is proved that γ(Λn) is bounded below by ⌈ Ln−2n n−3 ⌉ , where Ln is the n-th Lucas number. The 2-packing number ρ of these cubes is also studied. It is proved that ρ(Γn) is bounded below by 2 blg nc 2 −1 and the exact values of ...

Among numerical sequences, the Fibonacci numbers Fn have achieved a kind of celebrity status. Indeed, Koshy gushingly refers to them as one of the “two shining stars in the vast array of integer sequences” [16, p. xi]. The second of Koshy’s “shining stars” is the Lucas numbers, a close relative of the Fibonacci numbers, about which we will say more below. The Fibonacci numbers are famous for po...