A360535 Analog of Rudin-Shapiro sum sequence A020986, based on counting patterns 00 instead of 11.

1, 2, 3, 4, 3, 4, 5, 6, 7, 6, 7, 8, 7, 8, 9, 10, 9, 10, 9, 8, 7, 8, 9, 10, 11, 10, 11, 12, 11, 12, 13, 14, 15, 14, 15, 16, 17, 16, 15, 14, 15, 14, 15, 16, 15, 16, 17, 18, 17, 18, 17, 16, 15, 16, 17, 18, 19, 18, 19, 20, 19, 20, 21, 22, 21, 22, 21, 20, 19, 20

Offset: 0

Author

Jeffrey Shallit, Feb 10 2023

Keywords

• nonn

Comments

a(n) = Sum_{i=0..n} (-1)^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 A020986, 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.

Examples

			For n = 4, a(n) = 1+1+1+1+(-1) = 3.
		

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, A360536.