Generalized Fibonacci Sequences Research Articles

The exponential matrix: an explicit formula by an elementary method

We show an explicit formula, with a quite easy deduction, for the exponential matrix $e^{tA}$ of a real and finite square matrix $A$ (and for complex ones also). The elementary method developed avoids Jordan canonical form, eigenvectors, resolution of any linear system, matrix inversion, polynomial interpolation, complex integration, functional analysis, and generalized Fibonacci sequences. The basic tools are the Cayley-Hamilton theorem and the method of partial fraction decomposition. Two examples are given. We also show that such method applies to algebraic operators on infinite dimensional real Banach spaces.

Phi-Bonacci Butterfly Stroke Numbers to Assess Self-Similarity in Elite Swimmers

A harmonically self-similar temporal partition, which turns out to be subtly exhibited by elite swimmers at middle distance pace, is formally defined for one of the most technically advanced swimming strokes—the butterfly. This partition relies on the generalized Fibonacci sequence and the golden ratio. Quantitative indices, named ϕ-bonacci butterfly stroke numbers, are proposed to assess such an aforementioned hidden time-harmonic and self-similar structure. An experimental validation on seven international-level swimmers and two national-level swimmers was included. The results of this paper accordingly extend the previous findings in the literature regarding human walking and running at a comfortable speed and front crawl swimming strokes at a middle/long distance pace.

On $k$-Fibonacci balancing and $k$-Fibonacci Lucas-balancing numbers

The balancing number $n$ and the balancer $r$ are solution of the Diophantine equation $$1+2+\cdots+(n-1) = (n+1)+(n+2)+\cdots+(n+r). $$ It is well known that if $n$ is balancing number, then $8n^2 + 1$ is a perfect square and its positive square root is called a Lucas-balancing number. For an integer $k\geq 2$, let $(F_n^{(k)})_n$ be the $k$-generalized Fibonacci sequence which starts with $0,\ldots,0,1,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. The purpose of this paper is to show that 1, 6930 are the only balancing numbers and 1, 3 are the only Lucas-balancing numbers which are a term of $k$-generalized Fibonacci sequence. This generalizes the result from [Fibonacci Quart. 2004, 42 (4), 330-340].

Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑nk=0 xkWmk+j

In this paper, closed forms of the summation formulas ∑nk=0 xkWmk+j for generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Moreover, we give some identities and recurrence properties of generalized Fibonacci sequence.

A note on Q-matrices and higher order Fibonacci polynomials

The results described in a recent article, relative to a representation formula for the generalized Fibonacci sequences in terms of Q-matrices are extended to the case of Fibonacci, Tribonacci and R-bonacci polynomials.

Front crawl stroke in swimming: Phase durations and self-similarity

Human movements, such as walking and running, are able to generate rhythmic motor patterns, with the consequent appearance of hidden time-harmonic structures. Such harmonic structures are represented (at comfortable speed) by the occurrence of the golden ratio as ratio of durations of specific walking and running gait sub-phases. Preliminary experimental evidences suggest that front crawl swimming may behave, under this point of view, like walking and running. This paper aims to demonstrate that a mathematical connection between the golden ratio and the front crawl swimming stroke actually exists, at a pace that plays the role of the comfortable speed in walking and running. Generalized Fibonacci sequences are used to this purpose. They rely on the durations of aggregate phases of the front crawl swimming stroke with a clear physical meaning, while characterizing self-similarity of front crawl strokes in its simple nature and enhanced (stronger) variant. Experimental data on front crawl swimmers illustrate the theoretical derivations, suggesting that the pace playing the role of the comfortable speed in walking and running is the middle/long-distance one, while showing that the self-similarity level increases with the swimming technique and the enhanced self-similarity is associated with the performance of top-level swimmers.

Initial number of Lucas’ type series for the generalized Fibonacci sequence

Initial numbers for Lucas? type series have so far been established only for Fibonacci (2,1) and Tribonacci (3,1,3) sequences. Characteristics of stated series is their asymptotic relation with the exponent of the series constant. By using a simple procedure based on asymptotic relations of exponents of a sequences constant and Lucas? type series with the application of Nearest Integer Function - NIF, a general rule for initial numbers of Lucas? type series of Generalized Fibonacci sequence has been established, for the first time. All the gained initial numbers are integers, first initial number is always equal to the order of the sequence Fn(0) = n and remaining are functionally dependent on order of the number and are equal to Fn(k) = 2k-1-1. This is premiere presentation of Prim-nacci sequence, too. Determinants of initial numbers of the Lucas? type series for the generalized Fibonacci sequences are a proven factorial function.

A NEW GENERALIZED FIBONACCI SEQUENCE AND ITS PROPERTIES

The generalized Fibonacci sequence is defined as Fk = pFk-1 + qFk-2; k ≥2 with F0 = a; F1 = b where p, q, a, b all are positive integers. Panwar et al. derived the sequence {Vk}k≥0 which is generated by taking p = 1; q = a = b = 2 then Vk = Vk-1+ 2Vk-2; k ≥2 with V0= 2; V1 = 2 and {Uk}k≥0 which is generated by taking p = 1; q = a = 2; b = 0 then Uk = Uk-1+ 2Uk-2; k ≥2 with U0= 2; U1 = 0. The generalization of the Fibonacci sequence can be done in many ways by changing the initial condition and others by changing the recurrence relation. We introduce the more general sequence {Sk}k≥0 by taking p = 1, q = b = n, a = 0 then we have Sk = Sk-1+ nSk-2; k ≥2 with S0= 2; S1 = 0

Girard-Waring Type Formula for a Generalized Fibonacci Sequence

Girard-Waring Type Formula for a Generalized Fibonacci Sequence

An interesting generalized Fibonacci sequence: a two-by-two matrix representation

The Fibonacci sequence is a well-known example of second order recurrence sequence, which belongs to a particular class of recursive sequences. In this article, other generalized Fibonacci sequence is introduced and defined by $$\begin{aligned} F_{n+2}(\mu )=2aF_{n+1}(\mu )+(b-a^{2})F_{n}(\mu ),\ \ n\ge 0, \end{aligned}$$ where $$F_{0}(\mu )=-\mu $$ , $$F_{1}(\mu )=a^{2}-b-2a\mu $$ and $$\mu $$ is a real number. Also n-th power of the generating matrix for this generalized Fibonacci sequence is established and some basic properties of this sequence are obtained by matrix methods.

Generalized Fibonacci Sequences in Pythagorean Triple Preserving Matrices

Generalized Fibonacci Sequences in Pythagorean Triple Preserving Matrices

Ultra-wide unidirectional infrared absorber based on 1D gyromagnetic photonic crystals concatenated with general Fibonacci quasi-periodic structure in transverse magnetization

In this paper, an ultra-wide multiband unidirectional absorber based on 1D gyromagnetic magnetized photonic crystals (GMPCs) comprising ferrite material and isotropic dielectrics is investigated under transverse magnetization for transverse electric mode within the infrared regime by the transfer matrix method. Due to the Voigt magneto-optical effect, the magnetically modulated absorber possesses multiple preeminent one-way absorption bands, which run at 12.24–18.80, 25.25–50.05, 51.13–74.90 and 76.67–93.88 THz and in the reverse propagating direction. The absorber displays a splendid reflection phenomenon. These perfect nonreciprocal results can be ascribed to the destruction of the space-time reversal symmetry on account of the anisotropic gyromagnetic ferrite material and quasi-periodic arrangement of the proposed GMPCs. The comparisons of the absorption capacity among the general Fibonacci sequence, the periodic sequence and Thue–Morse sequence are discussed, and the general Fibonacci sequence has its superiority in the expansion of the one-way absorption bandwidth due to its better self-similarity. Besides, the regulative effects of the magnetic field intensity, damping factor and thicknesses of materials on the one-way absorption features are analyzed specifically. This research provides a promising design approach to the tunable infrared optical isolator, waveguide and circulator through the magnetized materials.

On the problem of Pillai with k-generalized Fibonacci numbers and powers of 3

For an integer [Formula: see text], let [Formula: see text] be the [Formula: see text]-generalized Fibonacci sequence which starts with [Formula: see text] (a total of [Formula: see text] terms) and for which each term afterwards is the sum of the [Formula: see text] preceding terms. In this paper, we find all integers [Formula: see text] with at least two representations as a difference between a [Formula: see text]-generalized Fibonacci number and a power of [Formula: see text]. This paper continues the previous work of the first author for the Fibonacci numbers, and for the Tribonacci numbers.

Complete generalized Fibonacci sequences modulo primes

We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence $\{F_n\}_{n=0}^\infty$ misses a congruence class modulo every prime large enough. On the other hand, if $Q \neq \pm 1$, we prove that (under GRH) the sequence $\{F_n\}_{n=0}^\infty$ hits every congruence class modulo infinitely many primes.

Gelin-Cesaro identities for generalized Fibonacci sequences

Gelin-Cesaro identities for generalized Fibonacci sequences

On Fibonacci Numbers and its Applications

In this article, we explore the representation of the product of k consecutive Fibonacci numbers as the sum of kth power of Fibonacci numbers. We also present a formula for finding the coefficients of the Fibonacci numbers appearing in this representation. Finally, we extend the idea to the case of generalized Fibonacci sequence and also, we produce another formula for finding the coefficients of Fibonacci numbers appearing in the representation of three consecutive Fibonacci numbers as a particular case. Also, we point out some amazing applications of Fibonacci numbers.

On the x–coordinates of Pell equations which are k–generalized Fibonacci numbers

For an integer k≥2, let {Fn(k)}n≥2−k be the k–generalized Fibonacci sequence which starts with 0,…,0,1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper, for an integer d≥2 which is square-free, we show that there is at most one value of the positive integer x participating in the Pell equation x2−dy2=±1, which is a k–generalized Fibonacci number, with a couple of parametric exceptions which we completely characterize. This paper extends previous work from [18] for the case k=2 and [17] for the case k=3.

Output multichannel optical filter based on hybrid photonic quasicrystals containing a high-Tc superconductor

In this study, we determined the properties of one-dimensional photonic quasicrystals (PQC), including superconductors and dielectric materials, based on two representative quasiperiodic sequences comprising the generalized Thue–Morse (GTM) and generalized Fibonacci sequences in order to design a multichannel optical filter. By using the transfer matrix method and two-fluid model, we determined that the transmittance spectra exhibited a series of photonic band gaps (PBGs). In order to improve the characteristics of the PBGs, we considered the GTM configuration and applied a deformation comprising y=xh+1 along the PQC device, where h represents the deformation degree, and x and y are the thicknesses of the layers before and after deformation, respectively.

On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes

The k-generalized Fibonacci sequence ( F n ( k ) ) n (sometimes also called k-bonacci or k-step Fibonacci sequence), with k ≥ 2 , is defined by the values 0 , 0 , … , 0 , 1 of starting k its terms and such way that each term afterwards is the sum of the k preceding terms. This paper is devoted to the proof of the fact that the Diophantine equation F m ( k ) = m t , with t > 1 and m > k + 1 , has only solutions F 12 ( 2 ) = 12 2 and F 9 ( 3 ) = 9 2 .

Number Sequence Prediction Problems for Evaluating Computational Powers of Neural Networks

Inspired by number series tests to measure human intelligence, we suggest number sequence prediction tasks to assess neural network models’ computational powers for solving algorithmic problems. We define the complexity and difficulty of a number sequence prediction task with the structure of the smallest automaton that can generate the sequence. We suggest two types of number sequence prediction problems: the number-level and the digit-level problems. The number-level problems format sequences as 2-dimensional grids of digits and the digit-level problems provide a single digit input per a time step. The complexity of a number-level sequence prediction can be defined with the depth of an equivalent combinatorial logic, and the complexity of a digit-level sequence prediction can be defined with an equivalent state automaton for the generation rule. Experiments with number-level sequences suggest that CNN models are capable of learning the compound operations of sequence generation rules, but the depths of the compound operations are limited. For the digitlevel problems, simple GRU and LSTM models can solve some problems with the complexity of finite state automata. Memory augmented models such as Stack-RNN, Attention, and Neural Turing Machines can solve the reverse-order task which has the complexity of simple pushdown automaton. However, all of above cannot solve general Fibonacci, Arithmetic or Geometric sequence generation problems that represent the complexity of queue automata or Turing machines. The results show that our number sequence prediction problems effectively evaluate machine learning models’ computational capabilities.