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March 1978 Über summen von Rudin-Shapiroschen koeffizienten
John Brillhart, Patrick Morton
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Illinois J. Math. 22(1): 126-148 (March 1978). DOI: 10.1215/ijm/1256048841

Abstract

The Rudin-Shapiro coefficients $\{a(n)\}$ are an infinite sequence of $\pm 1$'s, defined recursively by $a(0)=1$, $a(2n)=a(n)$, and $a(2n + 1)=(-1)^{n}a(n)$, $n \geq 0$ Various formulas are developed for the $n$th partial sum $s(n)$ and the $n$th alternating partial sum $t(n)$ of this sequence. These formulas are then used to show that $\sqrt{3/5} < s(n)/\surd n < \surd 6$ and $0 \leq; t(n)/\surd n < \surd 3$, $n \geq 1$ where the inequalities are sharp and the ratios are dense in the two intervals. For a given $n \geq 1$, the equation $s(k)= n$ is shown to have exactly $n$ solutions $k$.

Citation

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John Brillhart. Patrick Morton. "Über summen von Rudin-Shapiroschen koeffizienten." Illinois J. Math. 22 (1) 126 - 148, March 1978. https://doi.org/10.1215/ijm/1256048841

Information

Published: March 1978
First available in Project Euclid: 20 October 2009

zbMATH: 0371.10009
MathSciNet: MR0476686
Digital Object Identifier: 10.1215/ijm/1256048841

Subjects:
Primary: 10L10

Rights: Copyright © 1978 University of Illinois at Urbana-Champaign

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Vol.22 • No. 1 • March 1978
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