Counting powers of words in monoids

For a monoid M with presentation M = 〈 a 1 , … , a r | w 1 = w 2 , … , w 2 s − 1 = w 2 s 〉 , we count the number of words equivalent to w 1 n , n ∈ N , where equivalent means under the transitive closure of the relation generated by replacing an occurrence of w 2 i − 1 by w 2 i or vice versa (for any i ). Many interesting sequences are obtained in this way including the Fibonacci numbers.