On a Golay-Shapiro-Like Sequence

Author:

Allouche Jean-Paul1

Affiliation:

1. CNRS, Institut de Mathématiques, de Jussieu-PRG, Université Pierre et Marie Curie, Case 247, 4 Place Jussieu, F-75252 Paris Cedex 05, France

Abstract

Abstract 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.

Publisher

Walter de Gruyter GmbH

Reference9 articles.

1. [1] ALLOUCHE, J.-P.—LIARDET, P.: Generalized Rudin-Shapiro sequences, Acta Arith. 60 (1991) 1–27.

2. [2] BRILLHART, J.—CARLITZ, L.: Note on the Shapiro polynomials, Proc. Amer. Math. Soc. 25 (1970) 114–118.

3. [3] BRILLHART, J.—ERDŐS, P.—MORTON, P.: On sums of Rudin-Shapiro coefficients, II, Pacific J. Math. 107 (1983) 39–69.

4. [4] BRILLHART, J.—MORTON, P.: Über Summen von Rudin-Shapiroschen Koeffizienten, Illinois J. Math. 22 (1978) 126–148.

5. [5] BRILLHART, J.—MORTON, P.: A case study in mathematical research: The Golay-Rudin-Shapiro sequence, Amer. Math. Monthly 103 (1996) 854–869.

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