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

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 $\{{\mathsf{Z}}_0,{\mathsf{Z}}_1,{\mathsf{Z}}_2,{\mathsf{Z}}_n={\mathsf{Z}}_{n-3}+{\mathsf{Z}}_{n-2}+{\mathsf{Z}}_{n-1},n=3,\dots \}$, 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.