Oscillation properties of expected stopping times and stopping probabilities for patterns consisting of consecutive states in Markov chains

We investigate a Markov chain with a state space 1,2,…,r1,2,…,r stopping at appearance of patterns consisting of two consecutive states. It is observed that the expected stopping times of the chain have surprising oscillating dependencies on starting positions. Analogously, the stopping probabilities also have oscillating dependencies on terminal states. In a nonstopping Markov chain the frequencies of appearances of two consecutive states are found explicitly.