Generalized Fibonacci Sequences Research Articles

Generalized Fibonacci sequences in groupoids

Abstract In this paper, we introduce the notion of generalized Fibonacci sequences over a groupoid and discuss it in particular for the case where the groupoid contains idempotents and pre-idempotents. Using the notion of Smarandache-type P-algebra, we obtain several relations on groupoids which are derived from generalized Fibonacci sequences. MSC:11B39, 20N02, 06F35.

Coincidences in generalized Fibonacci sequences

The k -generalized Fibonacci sequence ( F n ( k ) ) n resembles the Fibonacci sequence in that it starts with 0 , … , 0 , 1 ( k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we are interested in finding the integers that appear in different generalized Fibonacci sequences; i.e., we study the Diophantine equation F n ( k ) = F m ( ℓ ) in positive integers n , k , m , ℓ with k , ℓ ⩾ 2 .

Generalized Fibonacci Sequence and its Properties

One of the simplest and more studied integer sequences is the Fibonacci sequence. Fibonacci sequence is famous for possessing wonderful and amazing properties. The Fibonacci sequence has been generalized in many ways, some by preserving the initial conditions and others by preserving the recurrence relation. In this paper, generalized Fibonacci sequence is introduced and defined by

Об арифметических свойствах обобщенной последовательности Фибоначчи и их следствиях

Об арифметических свойствах обобщенной последовательности Фибоначчи и их следствиях

A Matrix Approach for Divisibility Properties of the Generalized Fibonacci Sequence

We give divisibility properties of the generalized Fibonacci sequence by matrix methods. We also present new recursive identities for the generalized Fibonacci and Lucas sequences.

On waiting time distribution of runs of ones or zeros in a Bernoulli sequence

Consider an infinite sequence of Bernoulli trials {Xi|i=1,2,…}. Let W(k) denote the waiting time, the number of trials needed, to get either consecutive k ones or k zeros for the first time. The probability distribution of W(k) is derived for both independent and homogeneous two-state Markovian Bernoulli trials, using a generalized Fibonacci sequence of order k. For independent Bernoulli trials, a special case of symmetric trial with p=12 is considered.

A new narrowband multilayer DWDM optical filter in the order of defected Fibonacci sequence

The self-similar narrow stop-bands in the reflectivity spectrum of the structures in the order of generalized Fibonacci sequence are utilized in this study. A multilayer structure in the order of generalized Fibonacci sequence is defected by a proper layer to get narrow stop-bands identical to channels of dense wavelength division multiplexing (DWDM) filter with ITU-T standard (100 GHz grid). The index profile of the defected structure is apodized and index-matched to eliminate the sidelobes between the stop-bands. After these modifications, a thin, new DWDM filter is designed having significantly suppressed sidelobes.

Generalized Fibonacci sequences via orthogonal polynomials

Using tools from the theory of orthogonal polynomials we obtain and extend some recent results on generalized Fibonacci sequences, by stating a Binet’s-type formula for a sequence of real or complex numbers {Qn}n=0∞ defined byQ0=0,Q1=1,Qm=ajQm-1+bjQm-2,m≡j(modk),where k⩾3 is a fix integer number, and a0,a1,…,ak-1 and b0,b1,…,bk-1 are 2k given real or complex numbers, with bj≠0 for 0⩽j⩽k-1.

Generalized Fibonacci Sequences Generated from a $k$--Fibonacci Sequence

In this paper we will prove that all $k$--Fibonacci sequence contains generalized Fibonacci sequences and we will indicate the form of obtain them.

Fractal spectra in generalized Fibonacci one-dimensional magnonic quasicrystals

In this work we carry out a theoretical analysis of the spectra of magnons in quasiperiodic magnonic crystals arranged in accordance with generalized Fibonacci sequences in the exchange regime, by using a model based on a transfer-matrix method together random-phase approximation (RPA). The generalized Fibonacci sequences are characterized by an irrational parameter σ(p,q), which rules the physical properties of the system. We discussed the magnonic fractal spectra for first three generalizations, i.e., silver, bronze and nickel mean. By varying the generation number, we have found that the fragmentation process of allowed bands makes possible the emergence of new allowed magnonic bulk bands in spectra regions that were magnonic band gaps before, such as which occurs in doped semiconductor devices. This interesting property arises in one-dimensional magnonic quasicrystals fabricated in accordance to quasiperiodic sequences, without the need to introduce some deferent atomic layer or defect in the system. We also make a qualitative and quantitative investigations on these magnonic spectra by analyzing the distribution and magnitude of allowed bulk bands in function of the generalized Fibonacci number Fn and as well as how they scale as a function of the number of generations of the sequences, respectively.

Some Connections Between a Generalized Tribonacci Triangle and a Generalized Fibonacci Sequence

Some Connections Between a Generalized Tribonacci Triangle and a Generalized Fibonacci Sequence

On the intersection of two distinct $k$-generalized Fibonacci sequences

Let $k\geq 2$ and define $F^{(k)}:=(F_n^{(k)})_{n\geq 0}$, the $k$-generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_n^{(k)}=F_{n-1}^{(k)}+F_{n-2}^{(k)}+\cdots + F_{n-k}^{(k)}$, with initial conditions $0,0,\dots ,0,1$ ($k$ terms) and such that the first nonzero term is $F_1^{(k)}=1$. The sequences $F:=F^{(2)}$ and $T:=F^{(3)}$ are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation $F_n^{(k)}=F_m^{(\ell )}$. In this note, we use transcendental tools to provide a general method for finding the intersections $F^{(k)}\cap F^{(m)}$ which gives evidence supporting the Noe-Post conjecture. In particular, we prove that $F\cap T=\{0,1,2,13\}$.

The Research of the Generalized Fibonacci Sequence -based Propagation

Abstract T his paper assumed a kind of the propagated phenomenon of flow and load capacity, and tried to find a critical value through the generalized Fibonacci series-based and numerical simulation, which can explain when this node may lead to system breakdown.

Accelerations of Generalized Fibonacci Sequences

Accelerations of Generalized Fibonacci Sequences

A note on generalized Fibonacci sequences

Consider the generalized Fibonacci sequence {qn}n=0∞ having initial conditions q0=0,q1=1 and recurrence relation qn=aqn-1+qn-2 (when n is even) or qn=bqn-1+qn-2 (when n is odd), where a and b are nonzero real numbers. These sequences arise in a natural way in the study of continued fractions of quadratic irrationals and combinatorics on words or dynamical system theory. Some well-known sequences are special cases of this generalization. The Fibonacci sequence is a special case of {qn} with a=b=1. Pell’s sequence is {qn} with a=b=2 and the k-Fibonacci sequence is {qn} with a=b=k. In this article, we study numerous new properties of these sequences and investigate a sequence closely related to these sequences which can be regarded as a generalization of Lucas sequence of the first kind.

The generating function of a family of the sequences in terms of the continuant

Let a0,a1,…,ar−1 be positive numbers and define a sequence {qm}, with initial conditions q0=0 and q1=1, and for all m⩾2, qm=atqm−1+qm−2 where m≡t(modr). For r=2, the author called the sequence {qm} as the generalized Fibonacci sequences and studied it in [1]. But, it remains open to find a closed form of the generating function for general {qm}. In this paper, we solve this open problem, that is, we find a closed form of the generating function for {qm}in terms of the continuant.

Designing of stop band filters using hybrid periodic/quasi‐periodic one‐dimensional photonic crystals in microwave domain

AbstractUsing the transfer matrix method approach (TMM), the present paper attempts to determine the transmission properties of generalized quasi‐periodic (GQP) photonic systems. In addition, it studies hybrid periodic photonic/generalized quasi‐periodic photonic (PP/GQP) multilayer structures constructed as alternation of high‐low contrast (Roger and Air) of refractive indices. The quasi‐periodic photonic crystals (PCs) used in these hybrid hetero‐structures are the generalized Fibonacci sequence GF (m, n) and the generalized Thue–Morse sequence GTM (m, n) (m and n are integers). The results are presented in microwave domain for normal incident wave. By varying the parameters m and n of the quasi‐periodic structures, the corresponding transmission spectrum exhibits interesting properties.

On convergence of a new secant-like method for solving nonlinear equations

In this paper, we prove that the order of a new secant-like method presented recently with the so-called order of 2.618 is only 2.414. Some mistakes in the derivation leading to such a conclusion are pointed out. Meanwhile, under the assumption that the second derivative of the involved function is bounded, the convergence radius of the secant-like method is given, and error estimates matching its convergence order are also provided by using a generalized Fibonacci sequence.

Generalized Fibonacci sequences related to the extended hecke groups and an application to the extended modular group

The extended Hecke groups \overline{H}(\lambda _{q}) are generated by T(z)=-1/z, S(z)=-1/(z+\lambda _{q}) and R(z)=1/ \overline{z} with \lambda _{q}=2\cos (\pi /q) for q\geq 3 integer. In this paper, we obtain a sequence which is a generalized version of the Fibonacci sequence given in [6] for the extended modular group \overline{\Gamma }, in the extended Hecke groups \overline{H}(\lambda_{q}). Then we apply our results to \overline{\Gamma } to find all elements of the extended modular group \overline{\Gamma }.

Sing Generalized Fibonacci Sequences for Solving the One-Dimensional LQR Problem and its Discrete-Time Riccati Equation

In this article we develop a method of solving general one-dimensional Linear Quadratic Regulator (LQR) problems in optimal control theory, using a generalized form of Fibonacci numbers. We find th ...