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

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 700 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Diophantine Equation ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Sequence ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences

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 .

Generalized Fibonacci Sequences

2000 ◽  
Vol 93 (7) ◽  
pp. 604-606
Author(s):  
Sean Bradley
Keyword(s):  
Fibonacci Sequence ◽  
Infinite Family ◽  
Vast Amount ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences

Everyone loves the Fibonacci sequence. It is easy to describe, yet it gives rise to a vast amount of substantial mathematics. Physical applications and connections with various branches of mathematics abound. What could be better, unless someone told us that the Fibonacci sequence is but one member of an infinite family of sequences that we could be discussing? The generalization that follows has great potential for student and teacher exploration, as well as discovery, wonder, and amusement.

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

2021 ◽  
Vol 13 (1) ◽  
pp. 259-271
Author(s):  
S.E. Rihane
Keyword(s):  
Diophantine Equation ◽  
Fibonacci Sequence ◽  
Square Root ◽  
Generalized Fibonacci Sequence ◽  
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].

Generalization of the Fibonacci Sequence to n Dimensions

1966 ◽  
Vol 18 ◽  
pp. 332-349 ◽  
Author(s):  
George N. Raney
Keyword(s):  
Free Semigroup ◽  
Jacobi Polynomials ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Natural Generalization ◽  
Unimodular Group ◽  
Natural Extensions ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences

We introduce certain n X n matrices with integral elements that constitute a free semigroup with identity and generate the n-dimensional unimodular group. In terms of these matrices we define a certain sequence of n-dimensional vectors, which we show is the natural generalization to n dimensions of the Fibonacci sequence. Connections between the generalized Fibonacci sequences and certain Jacobi polynomials are found. The various basic identities concerning the Fibonacci numbers are shown to have natural extensions to n dimensions, and in some cases the proofs are rendered quite brief by the use of known theorems on matrices.

MULTIFRACTALITY OF GENERALIZED FIBONACCI PROFILES

Fractals ◽  
1993 ◽  
Vol 01 (03) ◽  
pp. 694-701 ◽  
Author(s):  
A.D. FREITAS ◽  
S. COUTINHO
Keyword(s):  
Ising Model ◽  
Initial Conditions ◽  
Fibonacci Sequence ◽  
Real Space ◽  
Local Magnetization ◽  
Renormalization Group Approach ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences ◽  
First Time ◽  
Infinite Set

The multifractal properties of a class of one parameter generalized Fibonacci sequences are studied. This class of recursion relations, which is defined by an infinite set of sequences similar to the original Fibonacci’s one, appears for the first time in the study of the Ising model by the real space Migdal-Kadanoff renormalization group approach. The whole set of numbers generated by these equations, when properly arranged over the interval [0,1], gives the exact local magnetization profile of the Ising model on generalized hierarchical lattices at the critical temperature. This profile has a multifractal structure showing that an infinite set of exponents is required to describe how its singularities are distributed. The F(α)-function for the measure defined by the normalized profile is numerically obtained and analyzed. Each value of α characterizes the set of numbers generated by one sequence with arbitrary initial conditions. The exact lower (αmin) and upper (αmax) bounds of the spectra are analytically calculated. For a particular value of the parameter, the original Fibonacci sequence appears, generating a set of numbers diverging with the αmin exponent.

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 O estudo de definições matemáticas no contexto de investigação histórica: um experimento didático envolvendo Engenharia Didática e Sequências Polinomiais de Fibonacci

2017 ◽  
Vol 6 (1) ◽  
Author(s):  
Rannyelly Rodrigues de Oliveira ◽  
Francisco Régis Vieira Alves ◽  
Rodrigo Sychocki da Silva
Keyword(s):  
History Of Mathematics ◽  
Fibonacci Sequence ◽  
Research Approach ◽  
Polynomial Sequence ◽  
Internal Validation ◽  
Generalized Fibonacci Sequence ◽  
Research Activities ◽  
Didactic Engineering ◽  
History Of ◽  
Fibonacci Polynomial

Resumo: O presente artigo apresenta uma abordagem de investigação no contexto da História da Matemática, envolvendo situações que visam oportunizar o entendimento da extensão, evolução e generalização de propriedades da Sequência de Fibonacci. Dessa forma, abordam-se duas situações. A primeira, envolvendo a descrição da fórmula de Binnet no campo dos inteiros. Logo em seguida, apresenta-se uma descrição e análise dos termos explícitos presentes na Sequência Polinomial de Fibonacci. O escopo da presente proposta de atividade busca a divulgação científica de noções envolvendo a generalização, ainda atual, fato que acentua o caráter ubíquo da Sequência de Fibonacci. À vista disso, a proposta de experimento didático está fundamentada na organização das características da Engenharia Didática. Almeja-se, além da validação interna das hipóteses levantadas durante a investigação, contribuir com a formação inicial de estudantes dos cursos de Licenciatura em Matemática que virem a estudar o tema.Palavras-chave: Atividades de investigação. Engenharia Didática. História da Matemática. Sequência Generalizada de Fibonacci.  THE STUDY OF MATHEMATICAL DEFINITIONS IN THE CONTEXT OF HISTORICAL RESEARCH: A DIDACTIC EXPERIMENT INVOLVING DIDACTIC ENGINEERING AND FIBONACCI POLYNOMIAL SEQUENCESAbstract: This article presents a research approach within the context of History of Mathematics, involving situations that aim to provide an understanding of the extension, evolution and generalization of properties of the Fibonacci Sequence. In this way, two situations are addressed. The first, involving the description of Binet's formula in the integer field. Then, a description and analysis of the explicit terms present in the Fibonacci Polynomial Sequence is presented. The scope of this activity proposal seeks the scientific dissemination of notions involving generalization, still current, a fact that accentuates the ubiquitous character of the Fibonacci Sequence. Thus the proposal of didactic experiment is based on the organized in the characteristics of Didactic Engineering, beyond the internal validation of the hypotheses raised during the investigation this paper aims at contributing to initial education of undergrad   Mathematicsof students that may come to study the subject.Keywords: Research activities. Didactic Engineering. History of Mathematics. Generalized Fibonacci Sequence.

scholarly journals On Fibonacci Numbers and its Applications

2019 ◽  
Vol 8 (12) ◽  
pp. 453-455
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Sequence

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.

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