Distribution Generated by a Random Inhomogenous Fibonacci Sequence

Let \(G_0=0\) $G_0=0$ and \(G_1=1\) $G_1=1$ . The present study deals with the inhomogeneous version \(\begin{aligned} G_n=G_{n-1}+G_{n-2}+w_{n-2} \end{aligned}\) $$\begin{array}{c}\hfill G_n=G_{n-1}+G_{n-2}+w_{n-2}\end{array}$$ of the Fibonacci sequence, where \(w_{n-2}\) $w_{n-2}$ takes value a with probability p, and does value b with \(1-p\) $1-p$ . We describe the probability distribution of the values of \(G_n\) $G_n$ with fixed n, and examine the properties like expected value and variance. The most challenging feature is the fractal-like structure of the distribution.