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

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