A restricted random walk defined via a Fibonacci process

In this article we study a random walk on a particularly simple graph. This walk is determined by a probabilistic process associated with the Fibonacci sequence. Exact formulas are derived for the expected proportions of time spent on each arc of the graph for a walk of length n, giving rise to sequences that do not appear in Sloane's On-Line Encyclopedia of Integer Sequences. We also obtain asymptotic relations for these expected proportions.