scholarly journals Pseudoprime values of the Fibonacci sequence, polynomials and the Euler function

2006 ◽  
Vol 17 (4) ◽  
pp. 611-625 ◽  
Author(s):  
Florian Luca ◽  
Igor E. Shparlinski
Keyword(s):  
Fibonacci Sequence ◽  
Euler Function

On the zero-multiplicity of the k-generalized Fibonacci sequence

2020 ◽  
Vol 26 (11-12) ◽  
pp. 1564-1578
Author(s):  
Jonathan García ◽  
Carlos A. Gómez ◽  
Florian Luca
Keyword(s):  
Fibonacci Sequence ◽  
Generalized Fibonacci Sequence

scholarly journals Note on some representations of general solutions to homogeneous linear difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Fibonacci Sequence ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
Homogeneous Linear

Abstract It is known that every solution to the second-order difference equation $x_{n}=x_{n-1}+x_{n-2}=0$ x n = x n − 1 + x n − 2 = 0 , $n\ge 2$ n ≥ 2 , can be written in the following form $x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ x n = x 0 f n − 1 + x 1 f n , where $f_{n}$ f n is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

scholarly journals Magnetic Normal Mode Calculations in Big Systems: A Highly Scalable Dynamical Matrix Approach Applied to a Fibonacci-Distorted Artificial Spin Ice

2021 ◽  
Vol 7 (3) ◽  
pp. 34
Author(s):  
Loris Giovannini ◽  
Barry W. Farmer ◽  
Justin S. Woods ◽  
Ali Frotanpour ◽  
Lance E. De Long ◽  
...  
Keyword(s):  
Normal Modes ◽  
Low Frequency ◽  
Fibonacci Sequence ◽  
Exchange Interactions ◽  
Geometric Distortion ◽  
Dynamical Matrix ◽  
Spin Ice ◽  
Magnetization Dynamics ◽  
Distortion Parameter ◽  
Artificial Spin Ice

We present a new formulation of the dynamical matrix method for computing the magnetic normal modes of a large system, resulting in a highly scalable approach. The motion equation, which takes into account external field, dipolar and ferromagnetic exchange interactions, is rewritten in the form of a generalized eigenvalue problem without any additional approximation. For its numerical implementation several solvers have been explored, along with preconditioning methods. This reformulation was conceived to extend the study of magnetization dynamics to a broader class of finer-mesh systems, such as three-dimensional, irregular or defective structures, which in recent times raised the interest among researchers. To test its effectiveness, we applied the method to investigate the magnetization dynamics of a hexagonal artificial spin-ice as a function of a geometric distortion parameter following the Fibonacci sequence. We found several important features characterizing the low frequency spin modes as the geometric distortion is gradually increased.

The Model for Icosahedral Al-Pd-Mn phase based on the T*(2F) Canonical Tiling

2000 ◽  
Vol 643 ◽  
Author(s):  
Gerald Kasner ◽  
Zorka Paradopolos
Keyword(s):  
Symmetry Axis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Fibonacci Sequence ◽  
Shortest Distance ◽  
Atomic Species ◽  
Three Dimensional Space

AbstractThe icosahedral canonical tiling of the three-dimensional space by six golden tetahedra T*(2F) [1] is decorated for physical applications by the Bergman polytopes [2]. The model can be also formulated as the “primitive) tiling TP [3] decorated by alternating Bergman symmetry axis of and icosahedron, there appear the plans on three mutual distances following the rule of a decorated Fibonacci sequence. All these three distances among the terraces (mutually scaled by a factor τ) have been recently observed by shen et al. [5]. In particular they have measured also the shortest distance of 2.52Å that breaks the Fibonnaci-sequence of terrace like surfaces measured previously by schaub et al. [6]. We predict the frequencies for the appearance of the terraces of different heights in the model under the condition that the model of Boudard et al. [7.8], we decorate the atomic positions by Al, Pd and Mn. We present images of the predicted possible terrace-like surfaces on three possible distances in the fully decorated model by the atomic species.

scholarly journals On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

2016 ◽  
Vol 67 (1) ◽  
pp. 41-46
Author(s):  
Pavel Trojovský
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Lucas Numbers ◽  
Initial Values ◽  
Lucas Sequence ◽  
Positive Integers

Abstract Let k ≥ 1 and denote (Fk,n)n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n−1 +Fk,n−2, with initial conditions Fk,0 = 0 and Fk,1 = 1. In the same way, the k-Lucas sequence (Lk,n)n≥0 is defined by satisfying the same recurrence relation with initial values Lk,0 = 2 and Lk,1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that Fk,n+1 + Fk,n−1 = Lk,n, for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and $F_{k,n + 1}^s + F_{k,n - 1}^s \in \left( {L_{k,m} } \right)_{m \ge 1} $ for infinitely many positive integers n, then s =1.

scholarly journals The Fibonacci Sequence: Nature’s Little Secret

2014 ◽  
Vol 2014 (1) ◽  
pp. 7-17 ◽  
Author(s):  
Nikoletta Minarova
Keyword(s):  
Fibonacci Sequence

Abstract Fibonacci: a natural design, easy to recognise - yet difficult to understand. Why do flowers and plants grow in such a way? It comes down to nature's sequential secret…This paper discusses how and when the Fibonacci sequence occurs in flora.

90.33 The r-subsequences of the Fibonacci sequence

2006 ◽  
Vol 90 (518) ◽  
pp. 263-266 ◽  
Author(s):  
John R. Silvester
Keyword(s):  
Fibonacci Sequence
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