Some properties of a Rudin–Shapiro-like sequence

We introduce the sequence (in)n≥0(in)n≥0 defined by in=(−1)inv2(n), where inv2(n)inv2(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)S(N) denotes the N  -th partial sum of the sequence (in)n≥0(in)n≥0, we show that S(N)=G(log4⁡N)N, where G   is a certain function that oscillates periodically between 3/3 and 2.