A360536 Analog of Rudin-Shapiro sum sequence A020990, based on counting patterns 00 instead of 11.

1, 0, 1, 0, -1, -2, -1, -2, -1, 0, 1, 0, -1, -2, -1, -2, -3, -4, -5, -4, -5, -6, -5, -6, -5, -4, -3, -4, -5, -6, -5, -6, -5, -4, -3, -4, -3, -2, -3, -2, -1, 0, 1, 0, -1, -2, -1, -2, -3, -4, -5, -4, -5, -6, -5, -6, -5, -4, -3, -4, -5, -6, -5, -6, -7, -8, -9

Offset: 0

Author

Jeffrey Shallit, Feb 10 2023

Keywords

• sign

Comments

a(n) = Sum_{i=0..n} (-1)^(i+e(i)), where e(i) counts the (possibly overlapping) occurrences of 00 in the base-2 representation of n. Note that e(0) = 0. This is the analog of A020990, which is the same sum, but with e(n) replaced by the function that counts the (possibly overlapping) occurrences of 11 in the base-2 representation of n.

Links

• N. Rampersad and J. Shallit, Rudin-Shapiro sums via automata theory and logic, Arxiv preprint arXiv:2302.00405 [math.NT], February 1 2023.

Crossrefs

Cf. A020985, A020990, A360535.