On a Golay-Shapiro-Like Sequence

A recent paper by P. Lafrance, N. Rampersad, and R. Yee studies the sequence of occurrences of 10 as a scattered subsequence in the binary expansion of integers. They prove in particular that the summatory function of this sequence has the “root N” property, analogously to the summatory function of the Golay-Shapiro sequence. We prove here that the root N property does not hold if we twist the sequence by powers of a complex number of modulus one, hence showing a fundamental difference with the Golay-Shapiro sequence.