A330222 Maximum autocorrelation of the first 2^n terms of the Rudin-Shapiro sequence A020985.

1, 1, 3, 5, 7, 13, 19, 33, 53, 85, 153, 217, 373, 557, 961, 1717, 2445

Offset

1,3

Comments

The j-th autocorrelation of the first m terms of a sequence r taking values in {1, -1} is defined the absolute value of the Sum_{0 <= i < m-j} r(i)*r(i+j). The maximum autocorrelation is the maximum of the absolute value of this quantity over the range 1 <= j < m. In our case r(i) = A020985(i) and n = 2^m.

Links

Table of n, a(n) for n=1..17.

T. Høholdt, H. Elbrønd Jensen, and J. Justesen, Aperiodic Correlations and the Merit Factor of a Class of Binary Sequences, IEEE Trans. Info. Theory IT-31 (1985), 549-552.

Formula

The paper of Høholdt et al. shows that a(n) = O( (2^n)^0.9 ).

Maple

b := (m, j) -> add(A020985(i)*A020985(i+j), i=0..m-j-1):
mb := m -> max(seq(abs(b(m, j)), j=1..m-1)):
a := n -> mb(2^n): seq(a(n), n=1..12); # Peter Luschny, Dec 06 2019

Crossrefs

Sequence in context: A084696 A352952 A354219 * A154700 A187872 A180450

Adjacent sequences: A330219 A330220 A330221 * A330223 A330224 A330225

Keyword

nonn,more

Author

Jeffrey Shallit, Dec 06 2019

Status

approved