Generalized Fibonacci Sequences Research Articles

On (k,p)-Fibonacci numbers and matrices

In this paper, some relations between the powers of any matrices $X$ satisfying the equation $X^k-pX^{k-1}-(p-1)X-I=\bf{0}$ and $(k,p)$-Fibonacci numbers are established with $k\geq2$. First, a result is obtained to find the powers of the matrices satisfying the condition above via $(k,p)$-Fibonacci numbers. Then, new properties related to $(k,p)$-Fibonacci numbers are given. Moreover, some relations between the sequence $\{F_{3,s}(n)\}$ and the generalized Fibonacci sequence $\{U_n(p,q)\}$ are also examined.

Dynamic continualization of mechanical metamaterials with quasi-periodic microstructure.

This article focuses on characterizing a class of quasi-periodic metamaterials created through the repeated arrangement of an elementary cell in a fixed direction. The elementary cell consists of two building blocks made of elastic materials and arranged according to the generalized Fibonacci sequence, giving rise to a quasi-periodic finite microstructure, also called Fibonacci generation. By exploiting the transfer matrix method, the frequency band structure of selected periodic approximants associated with the Fibonacci superlattice, i.e. the layered quasi-periodic metamaterial, is determined. The self-similarity of the frequency band structure is analysed by means of the invariants of the symplectic transfer matrix as well as the transmission coefficients of the finite clusters of Fibonacci generations. A high-frequency continualization scheme is then proposed to identify integral-type or gradient-type non-local continua. The frequency band structures obtained from the continualization scheme are compared with those derived from the Floquet-Bloch theory to validate the proposed scheme. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1).'

Cullen numbers and Woodall numbers in generalized Fibonacci sequences

Recently Bilu, Marques and Togbé [4] gave a general effective finiteness result on the equationFn(k)=Cm, where Fn(k) denotes the k-generalized Fibonacci-sequence and Cm the sequence of Cullen numbers, by giving explicit absolute bounds for n,k,m. However, the authors in [4] explained that their bounds were too large to use Dujella-Pethő reduction to completely solve the equation in question. In the present paper, using the bounds established by Bilu, Marques and Togbé in [4] and a different approach based on 2-adic analysis, we completely solve this equation. Further, using the same approach we also solve the corresponding equation for Woodall numbers.

On the Hankel matrices of finite rank and generalized Fibonacci sequences

On the Hankel matrices of finite rank and generalized Fibonacci sequences

Solving Skolem's problem for the k–generalized Fibonacci sequence with negative indices

The k–generalized Fibonacci sequence F(k) is the linearly recurrence sequence of order k whose first k terms are 0,…,0,1 and each term afterward is the sum of the preceding k terms. In this paper, we extend F(k) to negative indices obtaining a sequence denoted by H(k). Then, we solve Skolem's problem by finding all the zeros of H(k). To this end, we find very useful identities concerning H(k) similar to those obtained by others for F(k).

Generalized Fibonacci Sequences for Elliptic Curve Cryptography

The Fibonacci sequence is a well-known sequence of numbers with numerous applications in mathematics, computer science, and other fields. In recent years, there has been growing interest in studying Fibonacci-like sequences on elliptic curves. These sequences have a number of exciting properties and can be used to build new encryption systems. This paper presents a further generalization of the Fibonacci sequence defined on elliptic curves. We also describe an encryption system using this sequence which is based on the discrete logarithm problem on elliptic curves.

On Explicit Formulas of Hyperbolic Matrix Functions

Hyperbolic matrix functions are essential for solving hyperbolic coupled partial differential equations. In fact the best analytic-numerical approximations for resolving these equations come from the use of hyperbolic matrix functions. The hyperbolic matrix sine and cosine sh(A), ch(A) (A∈Mr(C)) can be calculated using numerous different techniques. In this article we derive some explicit formulas of sh(tA) and ch(tA) (t∈R) using the Fibonacci-H\"{o}rner and the polynomial decomposition, these decompositions are calculated using the generalized Fibonacci sequences combinatorial properties in the algebra of square matrices. Finally we introduce a third approach based on the homogeneous linear differential equations. And we provide some examples to illustrate your methods.

A note on generating primitive Pythagorean triples using matrices

We present matrices that generate families of primitive Pythagorean triples that arise from generalized Fibonacci sequences. We then present similar results for generalized Lucas sequences and primitive Pythagorean triples.

On Recurrences in Generalized Arithmetic Triangle

AbstractIn the present paper, we consider the generalized arithmetic triangle called GAT which is structurally identical to Pascal’s triangle for which we keep the Pascal’s rule of addition and we replace both legs by two sequences (an)n≥1and (bn)n≥1witha0=b0= Ω. Our goal is to describe the recurrence relation associated to the sum of elements lying along a finite ray in this triangle. As consequences, we obtain some combinatorial properties and we establish that the sum of elements lying along a main rising diagonal is a convolution of generalized Fibonacci sequence and another sequence which one will determine. We also precise the corresponding generating function. Further, we establish some nice identities by using the Morgan-Voyce phenomenon. Finally, we generalize the Golden ratio.

New method for solving non-homogeneous periodic second-order difference equation and some applications

In the present study, we are interested in solving the nonhomogeneous second-order linear difference equation with periodic coefficients of period pge 2, by bringing two new approaches enabling us to provide both analytic and combinatorial solutions to this family of equations. First, we get around the problem by converting this kind of equations to an equivalent family of nonhomogeneous linear difference equations of order p with constant coefficients. Second, we propose new expressions of the solutions of this family of equations, using our techniques of calculating the powers of product of companion matrices and some properties of generalized Fibonacci sequences. The study of the special case p=2 is provided. And to enhance the effectiveness of our approaches, some numerical examples are discussed.

Enhanced electron transport and self-similarity in quasiperiodic borophene nanoribbons with line defects.

Recent experiments have revealed multiple borophene phases of distinct lattice structures, suggesting that the unit cells of ν1/6 and ν1/5 boron sheets, namely α and β chains, serve as building blocks to assemble into novel borophene phases. Motivated by these experiments, we present a theoretical study of electron transport along two-terminal quasiperiodic borophene nanoribbons (BNRs), with the arrangement of the α and β chains following the generalized Fibonacci sequence. Our results indicate that the energy spectrum of these quasiperiodic BNRs is multifractal and characterized by numerous transmission peaks. In contrast to the Fibonacci model that all the electronic states should be critical, both delocalized and critical states appear in the quasiperiodic BNRs, where the averaged resistance saturates at the inverse of one conductance quantum for the delocalized states in the large length limit and contrarily exhibits a power-law dependence on the nanoribbon length for the critical states. Besides, the self-similarity is observed from the transmission spectrum, where the conductance curves overlap at different energy regions of two quasiperiodic BNRs of different Fibonacci indices and the resistance curves are analogous to each other at different energy scales of a single quasiperiodic BNR. These results complement previous studies on quasiperiodic systems where the multifractal energy spectrum and the self-similarity are observed by generating quasiperiodic potential energies, suggesting that borophene may provide an intriguing platform for understanding the structure-property relationships and exploring the physical properties of quasiperiodic systems.

On the reciprocal products of generalized Fibonacci sequences

In this paper, we use the properties of error estimation and the analytic method to study the reciprocal products of the bi-periodic Fibonacci sequence, the bi-periodic Lucas sequence, and the mth-order linear recursive sequence.

HOMOGENEOUS PLANS OF MULTI-FACTORY EXPERIMENTS ON QUASI-RANDOM R-ROBERTS SEQUENCES FOR SURROGATE MODELING IN A VORTEX STYLE STRUCTUROSCOPY

Context. The article is devoted to the creation of multifactorial experimental plans based on quasi-random recursive Roberts Rsequences. The object of the research is the process of creating computer-aided experimental design plans. The aim of the article is to create multifactorial, namely six- and seven-factor, uniform plans of experiments with low discrepancies, study of their projection properties and demonstrate their use on the example of surrogate modeling in eddy current structuroscopy.
 Method. An iterative method of unit hypercube even filling with reference points was used for constructing multidimensional experimental plans. It provides acceptable indicators of homogeneity and is realized on the basis of quasi-random nonparametric additive recursive Roberts R-sequences using irrational numbers, which, in turn, are obtained on the basis of the generalized Fibonacci sequence. The criterion for plans perfection is the assessment of homogeneity in terms of differences invariant with respect to the rotation of coordinates and re-marking and ordering of factors and which quantitatively characterize the deviation of the generated distribution from the ideal uniform.
 Results. Six- and seven-factor computer uniform experimental plans have been created for cataloging, which are characterized by low discrepancies and sufficiently high-quality projection properties. The tendency, which had been previously proved in the authors' research, for preserving these experimental plans characteristics in multidimensional factor spaces, which is observed with increasing number of plan points, has been confirmed. The evaluation of the quality of the created experimental plans is carried out both by visual analysis of the scattering matrix of all two-dimensional projections and by quantitative indicators of heterogeneity of the set of vectors that form the plan, namely centered and cyclic discrepancies.
 The example of the initial stage of creating a surrogate model to solve the problem of identifying profiles of electrophysical parameters in eddy current structuroscopy shows certain features of the application for created plans, in particular the transition from the plan for a unit hypercube to the plan in real factor space in the form of a hyperparallelepiped, which does not significantly affect its characteristics of homogeneity of the distribution of points.
 Conclusions. For the first time, the problem of creating six- and seven-factor uniform plans of experiments with low rates of centered and cyclic discrepancies based on R-sequences of Roberts was solved. The projection properties of the created experimental plans for different number of points were investigated. The method of constructing multidimensional computer plans of experiments taking into account the peculiarities of eddy current structuroscopy was improved. The use of six-dimensional experimental plans on the example of surrogate modeling in eddy current structuroscopy was demonstrated. The results of the study can be used in the construction of surrogate mathematical models of physical processes by any known methods of approximation.

Analysis of Limiting Ratios of Special Sequences

In this paper, we have determined the limit of ratio of (n+1)th term to the nth term of famous sequences in mathematics like Fibonacci Sequence, Fibonacci – Like Sequence, Pell's Sequence, Generalized Fibonacci Sequence, Padovan Sequence, Generalized Padovan Sequence, Narayana Sequence, Generalized Narayana Sequence, Generalized Recurrence Relations of Fibonacci – Type sequence, Polygonal Numbers, Catalan Sequence, Cayley numbers, Harmonic Numbers and Partition Numbers. We define this ratio as limiting ratio of the corresponding sequence. Sixteen different classes of special sequences are considered in this paper and we have determined the limiting ratios for each one of them. In particular, we have shown that the limiting ratios of Fibonacci sequence and Fibonacci – Like sequence is the fascinating real number called Golden Ratio which is 1.618 approximately. We have shown that the limiting ratio of Pell's sequence is a real number called Silver Ratio and the limiting ratios for generalized Fibonacci sequence are metallic ratios. We have also obtained the limiting ratios of Padovan and generalized Padovan sequence. The limiting ratio of Narayana sequence happens to be a number called super Golden Ratio which is 1.4655 approximately. We have shown that the limiting ratios of Generalized Narayana sequence are the numbers known as super Metallic Ratios. We have also shown that the limiting ratio of generalized recurrence relation of Fibonacci type is 2 and that of Polygonal numbers and Harmonic numbers are 1. We have proved that the limiting ratio of the famous Catalan sequence and Cayley numbers are 4. Finally, assuming Rademacher's Formula, we have shown that the limiting ratio of Partition numbers is the natural logarithmic base e. We have proved fourteen theorems to derive limiting ratios of various well known sequences in this paper. From these limiting ratio values, we can understand the asymptotic behavior of the terms of all these amusing sequences of numbers in mathematics. The limiting ratio values also provide an opportunity to apply in lots of counting and practical problems.

Explicit formulas for computing matrix trigonometric functions

Various methods have been introduced for calculating the matrix sine and cosine. This paper concerns the Fibonacci-Horner decomposition of the matrix sine and cosine sin(tA), cos(tA) (A ∈ M (r; ℂ), t ∈ ℝ), which is derived from the combinatorial properties of the generalized Fibonacci sequences in the algebra of square matrices. Finally we derive some formulas for the trigonometric functions of f(A) where f is an homographic function.

The Generalized Fibonacci Oscillator as an Open Quantum System

We consider an open quantum system with Hamiltonian $H_S$ whose spectrum is given by a generalized Fibonacci sequence weakly coupled to a Boson reservoir in equilibrium at inverse temperature $\beta$. We find the generator of the reduced system evolution and explicitly compute the stationary state of the system, that turns out to be unique and faithful, in terms of parameters of the model. If the system Hamiltonian is generic we show that convergence towards the invariant state is exponentially fast and compute explicitly the spectral gap for low temperatures, when quantum features of the system are more significant, under an additional assumption on the spectrum of $H_S$.

Some equalities and binomial sums about the generalized Fibonacci number $u_n$

In this study, we take the generalized Fibonacci sequence \{u_{n}\} as u_{0}=0,u_{1}=1 and \ u_{n}=ru_{n-1}+u_{n-2} for n>1, where r is a non-zero integer. Based on Halton’s paper in [4], we derive three interrelated functions involving the terms of generalized Fibonacci sequence \{u_{n}\}. Using these three functions we introduce a simple approach to obtain a lot of identities, binomial sums and alternate binomial sums involving the terms of generalized Fibonacci sequence \{u_{n}\}.

“Generating matrix for Generalized Fibonacci numbers and Fibonacci polynomials

Many researchers have been working on recurrence relation sequences of numbers and polynomials which are useful topic not only in mathematics but also in physics, economics and various applications in many other fields. There are many useful identities on recurrence relation sequence but there main problem to find any term of recurrence relation sequence we need to find previous all terms of recurrence relation sequence of numbers and polynomials. There were many important theorems obtained on recurrence relation sequences. In this paper we have given special identity for generalized Fibonacci sequence of number and Fibonacci sequence of polynomials. These identities are very useful to represent Fibonacci generalized sequence of numbers and Fibonacci sequence of polynomials in the form of matrix. Authors define a special formula in this paper by this we can find special representation of Fibonacci generalized sequence of numbers and Fibonacci sequence of polynomials in the form of matrix. So, we can say that this paper is generalization of property of Fibonacci sequence of number and Fibonacci sequence of polynomials

Polynomials Related to Generalized Fibonacci Sequence

In this paper we extend the Fibonacci recurrence relation to define the sequence {Cn } and derive some properties of this sequence. We also define the four comparison sequence {Pn }, {Qn }, {Rn }, {Sn }. We also obtain some identities with the help of generating matrix.

On the distribution of the roots of polynomial $z^k-z^{k-1}-\dots - z-1$

We consider the polynomial $f_k(z) = z^k-z^{k-1}-\cdots -z-1$ for $k\ge 2$ which arises as the characteristic polynomial of the $k$-generalized Fibonacci sequence. In this short paper, we give estimates for the absolute values of the roots of $f_k(z)$ which lie inside the unit disk.