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...

Various generalizations of the Fibonacci numbers have been proposed, studied and applied over the years (see [5] for a brief list). Probably the best known are the k-generalized Fibonacci numbers F (k) n (also known as the k-fold Fibonacci, k-th order Fibonacci, k-Fibonacci or polynacci numbers), satisfying F (k) n = F (k) n−1 + F (k) n−2 + . . .+ F (k) n−k , n ≥ k , F (k) n = 0 , 0 ≤ n ≤ k − 2...

A general formulation of discrete optimization is to maximize a given function f : D → R over a discrete (finite) domain D. In general, of course, this problem may require |D| probes to f . One approach to making optimization more tractable is to be satisfied with finding a local maximum, i.e., a point at which f attains a value larger than all “neighboring” points, for some definition of neigh...

The Fibonacci operator approach inspired by Andrews (2004) is explored to investigate q-analogs of the generalized Fibonacci and Lucas polynomials introduced by Chu and Vicenti (2003). Their generating functions are compactly expressed in terms of Fibonacci operator fractions. A determinant evaluation on q-binomial coefficients is also established which extends a recent result of Sun (2005).

We define the convolved hðxÞ-Fibonacci polynomials as an extension of the classical con-volved Fibonacci numbers. Then we give some combinatorial formulas involving the hðxÞ-Fibonacci and hðxÞ-Lucas polynomials. Moreover we obtain the convolved hðxÞ-Fibo-nacci polynomials from a family of Hessenberg matrices. Fibonacci numbers and their generalizations have many interesting properties and appli...

The paper presents a solution to the problem related reconstruction of parameters in mathematical model bacterial colony patterns domain with symmetry. inverse consists determining value diffusion coefficient active bacteria. describing distribution bacteria given region, as well concentration substrate over time is considered. Such system partial differential equations appropriate initial-boun...

For a finitely generated group G = 〈A〉 where A = {a1, a2, . . . , an} the sequence xi = ai+1, 0 ≤ i ≤ n − 1, xi+n = ∏n j=1 xi+j−1, i ≥ 0, is called the Fibonacci orbit of G with respect to the generating set A, denoted FA(G). If FA(G) is periodic we call the length of the period of the sequence the Fibonacci length of G with respect to A, written LENA(G). In this paper we examine the Fibonacci ...

The “Fibonacci Dichotomy” of Kaiser and Klazar [15] was one of the first general results on the enumeration of permutation classes. It states that if there are fewer permutations of length n in a permutation class than the nth Fibonacci number, for any n, then the enumeration of the class is given by a polynomial for sufficiently large n. Since the Fibonacci Dichotomy was established for permut...

In an integrated sensing and communication (ISAC) system, improving the accuracy of delay Doppler shift parameter estimation is a critical task that sustains performance system. To tackle this task, we introduce two-stage algorithm named matched filter-Fibonacci (MF-F), which employs orthogonal time frequency space (OTFS) waveform characteristics in delay-Doppler (DD) domain. first step (MF), p...