The m-th Cullen number Cm is a number of the form m2 m + 1 and the m-th Woodall number Wm has the form m2 m − 1. In 2003, Luca and Stănică proved that the largest Fibonacci number in the Cullen sequence is F4 = 3 and that F1 = F2 = 1 are the largest Fibonacci numbers in the Woodall sequence. Very recently, the second author proved that, for any given s > 1, the equation Fn = ms m ± 1 has only f...

In this paper, we suggest a lower and an upper bound for the Generalized Fibonaccip-Sequence, for different values of p. The Fibonacci-p-Sequence is a generalization of the Classical Fibonacci Sequence. We first show that the ratio of two consecutive terms in generalized Fibonacci sequence converges to a p-degree polynomial and then use this result to prove the bounds for generalized Fibonacci-...

Abstract— Identification whether a certain number belongs to either Fibonacci or Lucas sequence is computationally intense task in the sense that the sequence itself need to be written from the beginning until the requested number. In this paper we present a method of detecting whether a certain number is of Lucas or Fibonacci sequence as well as introduce relationships between such sequences. ...

In this paper, we define new kinds of Fibonacci numbers, which generalize both Fibonacci, Jacobsthal, Narayana numbers and p-numbers in the distance sense, using definition a between by recurrence relation according to parameter k. Tiling combinatorial interpretations these are presented, explicit formulas that allow us calculate nth number given. Also, their generating functions obtained sums ...

We continue our study of the class of Fibonacci-automatic words. These are infinite words whose nth term is defined in terms of a finite-state function of the Fibonacci representation of n. In this paper, we show how enumeration questions (such as counting the number of squares of length n in the Fibonacci word) can be decided purely mechanically, using a decision procedure. We reprove some kno...

Fibonacci numbers and difference equations show up in many counting problems. Zeckendorf showed how to represent natural numbers in “binary” Fibonacci bases. Capocelli counted the number of 0 bits and 1 bits in such representation. Here we use the theory of difference equations to try to provide proofs for Capocelli’s claims. We also investigate generalization of the Fibonacci difference equati...

Let denote by Fk,n the nth k-Fibonacci number where Fk,n = kFk,n−1+Fk,n−2 for n 2 with initial conditions Fk,0 = 0, Fk,1 = 1, we may derive a functionalequation f(k, x) = kf(k, x − 1) + f(k, x − 2). In this paper, we solve thisequation and prove its Hyere-Ulam stability in the class of functions f : N×R ! X,where X is a real Banach space.

The Fibonacci cube of dimension n, denoted as Γn, is the subgraph of n-cube Qn induced by vertices with no consecutive 1’s. We study the maximum number of disjoint subgraphs in Γn isomorphic to Qk, and denote this number by qk(n). We prove several recursive results for qk(n), in particular we prove that qk(n) = qk−1(n − 2) + qk(n − 3). We also prove a closed formula in which qk(n) is given in t...

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 ...