## Rudin-Shapiro Sequence

The sequence of numbers given by

 (1)

where is written in binary
 (2)

It is therefore the parity of the number of pairs of consecutive 1s in the Binary expansion of . The Summatory sequence is
 (3)

which gives
 (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.