Meta-automatic Sequences

Nested (or meta-Fibonacci) recurrences, such as the recurrence used to define Hofstadter's Q-sequence, along with the digit-based recurrences that underlie automatic sequences are of interest from both number-theoretic and combinatorial points of view. In this direction, Allouche and Shallit showed how the frequency sequence of a variant of the $Q$-sequence is $2$-automatic. This inspires us to introduce what may be seen as a natural combination of the recurrences for meta-Fibonacci and automatic sequences, by introducing the concept of a meta-automatic sequence. We exhibit two binary meta-automatic sequences $M_1$ and $M_2$ whose defining recurrences do not satisfy the Allouche-Shallit automaticity criterion directly, and this is formalized in our paper. For each of these integer sequences $M_1$ and $M_2$, we prove explicit DFAO evaluations, together with $4$-uniform morphisms, and we also consider the factor complexities of these sequences.