A Family of Sequences Generalizing the Thue Morse and Rudin Shapiro Sequences

For $m \ge 1,$ let $P_m =1^m,$ the binary string of $m$ ones. Further define the infinite sequence $s_m$ by $s_{m,n} = 1$ iff the number of (possibly overlapping) occurrences of $P_m$ in the binary representation of $n$ is odd, $n \ge 0.$ For $m=1,2$ respectively $s_m$ is the Thue-Morse and Rudin-Shapiro sequences. This paper shows that for each $m\ge 1,$ (i) $s_m$ is automatic; (ii) the minimal, DFA (deterministic finite automata) accepting $s_m$ has $2m$ states; (iii) it suffices to use prefixes of length $2^{m-1}$ to distinguish all sequences in the 2-kernel of $s_m$; and (iv) the characteristic function of the length $2^{m-1}$ prefix of the 2-kernel sequences of $s_m$ can be formulated using the Vile and Jacobsthal sequences. The proofs are based on a correspondence between binary strings under concatenation and integers under addition and multiplication. Both Mathematica and Walnut are employed for exploratory analysis of patterns. The paper presents results about maximal runs, palindromes, order of squares, and borders in $s_m,$ generalizing similar results for $s_1$ and $s_2.$ In the conclusion we suggest that families of automatic sequences is a fruitful concept and a useful group of sequences extending automatic sequences similar to the regular and synchronized sequences.