Counting Subwords Occurrences in Base-b Expansions

We consider the sequence (Sb(n))n≥0 counting the number of distinct (scattered) subwords occurring in the base-b expansion of the non-negative integers. By using a convenient tree structure, we provide recurrence relations for (Sb(n))n≥0 leading to the b-regularity of the latter sequence. Then we deduce the asymptotics of the summatory function of the sequence (Sb(n))n≥0.