A Recurrence Formula for the Fibonacci Sequence Generated from Rabbits with a Finite Lifespan

2018 ◽  
Vol 10 (1) ◽  
pp. 55-61
Author(s):  
Jooseok Noh ◽  
◽  
Yoora Kim ◽  
Keyword(s):  
Recurrence Formula ◽  
Fibonacci Sequence

A MINIMAL LATTICE MODEL RELATED TO PHYLLOTAXY

1996 ◽  
Vol 04 (02) ◽  
pp. 199-218
Author(s):  
X.D. FAN
Keyword(s):  
Growth Rate ◽  
Analytical Solution ◽  
Lattice Model ◽  
Lattice Theory ◽  
Initial Conditions ◽  
Recurrence Formula ◽  
Fibonacci Sequence ◽  
Divergence Angle ◽  
Asymptotic Values ◽  
Minimal Lattice

Structures in phyllotaxy are generalized using some concepts from lattice theory. A minimal lattice model is introduced to simulate the dynamics of the morphology of phyllotaxis. Numerical results indicate that the lattice divergences, which are equivalent to the divergence angle as used in phyllotaxy, converge to constant numbers for most of the growth rate and initial conditions. These numbers are associated with the noble numbers which are related to the general Fibonacci sequence. A method is developed to derive the lattice divergence as a sequence expressed in a recurrence formula. An analytical solution of the latter is obtained which reveals the convergence and asymptotic values of the sequence.

scholarly journals k-Step Sum and m-Step Gap Fibonacci Sequence

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Maria Adam ◽  
Nicholas Assimakis
Keyword(s):  
Spectral Radius ◽  
Recurrence Formula ◽  
Fibonacci Sequence

For two given integers k, m, we introduce the k-step sumand m-step gap Fibonacci sequence by presenting a recurrence formula that generates the nth term as the sum of k successive previous terms starting the sum at the mth previous term. Known sequences, like Fibonacci, tribonacci, tetranacci, and Padovan sequences, are derived for specific values of k, m. Two limiting properties concerning the terms of the sequence are presented. The limits are related to the spectral radius of the associated {0,1}-matrix.

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.

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