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...

The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from elementary properties of the Fibonacci numbers. 1 Preliminaries The well-known sequence of Fibonacci numbers (sequence A000045 in the On-Line Encyclopedia of Int...

If p = 1 then the resulting tree is the Fibonacci tree. It can easily be verified that the number of A’s in the n-th layer equals the n-th Fibonacci number Fn and the number of B’s equals Fn−1. Let An and Bn denote the number of A’s and B’s, respectively, in the first n layers of the tree. Then we have An An + Bn = ∑n i=1 Fi ∑n+1 i=2 Fi = 1 − Fn+1 − F1 ∑n+1 i=2 Fi Using the well known represent...

We study the Fibonacci sets from the point of view of their quality with respect to discrepancy and numerical integration. Let {bn}n=0 be the sequence of Fibonacci numbers. The bn-point Fibonacci set Fn ⊂ [0, 1] is defined as Fn := {(μ/bn, {μbn−1/bn})} μ=1, where {x} is the fractional part of a number x ∈ R. It is known that cubature formulas based on Fibonacci set Fn give optimal rate of error...

In this paper, we introduce a tiling approach to (p,q)-Fibonacci and (p,q)-Lucas numbers that generalize of the well-known Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal ve Jacobsthal-Lucas numbers. We show nth number is interpreted as ways tile 1×n board with cells labeled 1,2,...,n using colored 1×1 squares 1×2 dominoes, where there are p kind colors for q dominoes. Then circular also present...

The combinatorial properties of the Fibonacci innnite word are of great interest in some aspects of mathematics and physics, such as number theory, fractal geometry, formal language, computational complexity , quasicrystals etc. In this note, we introduce the singular words of the Fibonacci innnite word and discuss their properties. We establish two decompositions of the Fibonacci word in singu...