On the regularity of [formula omitted].

Let α , β be real numbers and b ≥ 2 be an integer. Allouche and Shallit showed that the sequence { ⌊ α n + β ⌋ } n ≥ 0 is b -regular if and only if α is rational. In this paper, using a base-independent regular language, we prove a similar result that the sequence { ⌊ log b ⁡ ( α n + β ) ⌋ } n ≥ 0 is b -regular if and only if α is rational, where α > 0 , β ≥ 0 . In particular, when α = 2 , β = 0 and b = 2 , we answer a question of Allouche and Shallit that the sequence { ⌊ 1 2 + log 2 ⁡ n ⌋ } n ≥ 1 is not 2-regular, which has previously been proved by Bell, Moshe and Rowland respectively. [ABSTRACT FROM AUTHOR]