Some remarks on recursive sequence of fibonacci type

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.