Abstract
Let {r(n)}n≥0 be the Rudin-Shapiro sequence, and let ρ(n):=max{∑j=ii+n−1r(j)∣i≥0}+1 be the abelian complexity function of the Rudin-Shapiro sequence. In this note, we show that the function ρ(n) has many similarities with the classical summatory function Sr(n):=∑i=0nr(i). In particular, we prove that for every positive integer n, 3≤ρ(n)n≤3. Moreover, the point set {ρ(n)n:n≥1} is dense in [3,3].
Highlights
0} + 1 be the abelian complexity function of the Rudin-Shapiro sequence
We show that the function ρ(n) has many similarities with the classical summatory function Sr (n) := ∑in=0 r (i )
Brillhart and Morton proved that for every n ≥ 1, Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
Summary
0} + 1 be the abelian complexity function of the Rudin-Shapiro sequence. It has been proved in [2] that the sequence ρ(n) satisfies ρ(1) = 2, ρ(2) = 3, ρ(3) = 4 and for every n ≥ 1, ρ(4n) = 2ρ(n) + 1, ρ(4n + 1) = 2ρ(n), ρ(4n + 2) = ρ(n) + ρ(n + 1), ρ(4n + 3) = 2ρ(n + 1). =i the abelian complexity ρ(n) of the Rudin-Shapiro sequence satisfies ρ(n) = Mr (n) + 1 for every n ≥ 1.
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