Probabilistic approaches to exploring Binet's type formula for the Tribonacci sequence

Skander Hachicha; Najmeddine Attia; Skander Hachicha; Najmeddine Attia

doi:10.3934/math.2025540

AIMS Mathematics

2025, Volume 10, Issue 5: 11957-11975. doi: 10.3934/math.2025540

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Probabilistic approaches to exploring Binet's type formula for the Tribonacci sequence

Skander Hachicha ,
Najmeddine Attia ^,

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

Received: 21 March 2025 Revised: 12 May 2025 Accepted: 15 May 2025 Published: 23 May 2025
MSC : 11B39, 60G50

This paper presents a detailed procedure for deriving a Binet's type formula for the Tribonacci sequence $ \{ {\mathsf T}_n\} $. We examine the limiting distribution of a Markov chain that encapsulates the entire sequence $ \{ {\mathsf T}_n\} $, offering insights into its asymptotic behavior. An approximation of $ {\mathsf T}_n $ is provided using two distinct probabilistic approaches. Furthermore, we study random sequences of the form $ \{ {\mathsf Z}_0, {\mathsf Z}_1, {\mathsf Z}_2, {\mathsf Z}_n = {\mathsf Z}_{n-3} + {\mathsf Z}_{n-2} + {\mathsf Z}_{n-1}, n = 3, \ldots\} $, referred to as the Tribonacci sequence of Random Variables. These sequences, fully defined by their initial random variables, are analyzed in terms of their distributional and limiting properties.
- Tribonacci sequence,
- Binet formula,
- Tribonacci random variable,
- Markov chain
Citation: Skander Hachicha, Najmeddine Attia. Probabilistic approaches to exploring Binet's type formula for the Tribonacci sequence[J]. AIMS Mathematics, 2025, 10(5): 11957-11975. doi: 10.3934/math.2025540

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Abstract

This paper presents a detailed procedure for deriving a Binet's type formula for the Tribonacci sequence $ \{ {\mathsf T}_n\} $. We examine the limiting distribution of a Markov chain that encapsulates the entire sequence $ \{ {\mathsf T}_n\} $, offering insights into its asymptotic behavior. An approximation of $ {\mathsf T}_n $ is provided using two distinct probabilistic approaches. Furthermore, we study random sequences of the form $ \{ {\mathsf Z}_0, {\mathsf Z}_1, {\mathsf Z}_2, {\mathsf Z}_n = {\mathsf Z}_{n-3} + {\mathsf Z}_{n-2} + {\mathsf Z}_{n-1}, n = 3, \ldots\} $, referred to as the Tribonacci sequence of Random Variables. These sequences, fully defined by their initial random variables, are analyzed in terms of their distributional and limiting properties.

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