scholarly journals A Matrix Approach for Divisibility Properties of the Generalized Fibonacci Sequence

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Aynur Yalçiner
Keyword(s):  
Fibonacci Sequence ◽  
Matrix Approach ◽  
Matrix Methods ◽  
Generalized Fibonacci Sequence ◽  
Lucas Sequences ◽  
Divisibility Properties

We give divisibility properties of the generalized Fibonacci sequence by matrix methods. We also present new recursive identities for the generalized Fibonacci and Lucas sequences.

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.

scholarly journals The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of z(n)

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2912
Author(s):  
Eva Trojovská ◽  
Venkatachalam Kandasamy
Keyword(s):  
Positive Integer ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Natural Density ◽  
Almost All ◽  
Divisibility Properties

Let (Fn)n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z(n)=min{k≥1:n∣Fk}. Very recently, Trojovská and Venkatachalam proved that, for any k≥1, the number z(n) is divisible by 2k, for almost all integers n≥1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2k by any integer m≥1. In this paper, in particular, we prove this conjecture.

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 Sequence and its Properties

2013 ◽  
Vol 1 (6) ◽  
pp. 194
Author(s):  
Bijendra Singh ◽  
Shikha Bhatnagar ◽  
Omprakash Sikhwal
Keyword(s):  
Fibonacci Sequence ◽  
Generalized Fibonacci Sequence
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