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, \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.