Rudin-Shapiro Sequence

The sequence of numbers given by

$$a_n=(-1)^{\sum_{i=1}^{k-1} \epsilon_i\epsilon_{i+1}},$$ (1)

where $n$ is written in binary

$$n=\epsilon_1 \epsilon_2 \ldots \epsilon_k.$$ (2)

It is therefore the parity of the number of pairs of consecutive 1s in the Binary expansion of $n$. The Summatory sequence is

$$s_n\equiv \sum_{j=0}^n a_j,$$ (3)

which gives

$$s_n=\cases{ 2^{k/2}+1 & if $k$\ is even\cr 2^{(k-1)/2}+1 & if $k$\ is odd\cr}$$ (4)

(Blecksmith and Laud 1995).

References

Blecksmith, R. and Laud, P. W. ``Some Exact Number Theory Computations via Probability Mechanisms.'' Amer. Math. Monthly 102, 893-903, 1995.

Brillhart, J.; Erdös, P.; and Morton, P. ``On the Sums of the Rudin-Shapiro Coefficients II.'' Pac. J. Math. 107, 39-69, 1983.

Brillhart, J. and Morton, P. ``Über Summen von Rudin-Shapiroschen Koeffizienten.'' Ill. J. Math. 22, 126-148, 1978.

France, M. M. and van der Poorten, A. J. ``Arithmetic and Analytic Properties of Paper Folding Sequences.'' Bull. Austral. Math. Soc. 24, 123-131, 1981.