Some remarks on recursive sequence of fibonacci type

Najmeddine Attia; Najmeddine Attia

doi:10.3934/math.20241262

AIMS Mathematics

2024, Volume 9, Issue 9: 25834-25848. doi: 10.3934/math.20241262

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Research article

Some remarks on recursive sequence of fibonacci type

Najmeddine Attia ^,

Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia

Received: 08 June 2024 Revised: 20 August 2024 Accepted: 22 August 2024 Published: 05 September 2024
MSC : 11B39, 60G50, 28A80

This paper presents a detailed procedure for determining the probability of return for random walks on $ \mathbb{Z} $, whose increment is given by a generalization of a well-known Fibonacci sequence, namely the $ k $-Fibonacci-like sequence $ (G_{k, n})_n $. Also, we study the size of the set of these walks that return to the origin an infinite number of times, in term of fractal dimension. In addition, we investigate the limiting distribution of an adequate Markov chain that encapsulates the entire Tribonacci sequence $ ({\mathsf T}_n) $ to provide the limiting behavior of this sequence.
- random walks,
- $ k $-Fibonacci like sequence,
- probability of return,
- fractal dimension,
- Markov chain,
- Binet formula
Citation: Najmeddine Attia. Some remarks on recursive sequence of fibonacci type[J]. AIMS Mathematics, 2024, 9(9): 25834-25848. doi: 10.3934/math.20241262

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Abstract

This paper presents a detailed procedure for determining the probability of return for random walks on $ \mathbb{Z} $, whose increment is given by a generalization of a well-known Fibonacci sequence, namely the $ k $-Fibonacci-like sequence $ (G_{k, n})_n $. Also, we study the size of the set of these walks that return to the origin an infinite number of times, in term of fractal dimension. In addition, we investigate the limiting distribution of an adequate Markov chain that encapsulates the entire Tribonacci sequence $ ({\mathsf T}_n) $ to provide the limiting behavior of this sequence.

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