Some properties of a Rudin-Shapiro-like sequence

Abstract

We introduce the sequence $(i_n{)}_{n\ge 0}$ defined by $i_n=(-1{)}^{in{v}_2(n)}$, where $in{v}_2(n)$ denotes the number of inversions (i.e., occurrences of 10 as a scattered subsequence) in the binary representation of n. We show that this sequence has many similarities to the classical Rudin-Shapiro sequence. In particular, if S(N) denotes the N-th partial sum of the sequence $(i_n{)}_{n\ge 0}$, we show that $S(N)=G({\log }_{4}N)\sqrt{N}$, where G is a certain function that oscillates periodically between $\sqrt{3}/3$ and $\sqrt{2}$.

arXiv comment21 pages, 6 figures