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A282339
A pseudorandom binary sequence with minimum variance of the absolute values of its discrete Fourier transform.
1
1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1
OFFSET
1,1
COMMENTS
a(1) = 1. Each subsequent term is chosen so as to minimize the variance of the absolute values of the discrete Fourier transform of the partial sequence. If the variance doesn't change with different choices for the next term, then the complement of the previous term is used. The algorithm works on a sequence of 1's and -1's then, as a last step, all -1's are replaced by 0's.
This sequence is similar to A282343 where the peak-to-peak distance is considered instead of the variance.
MATHEMATICA
varfourier[x_]:=Variance[Abs[Fourier[x]]];
a={1}; (*First element*)
nmax=120; (*number of appended elements*)
Do[If[varfourier[Append[a, 1]]<varfourier[Append[a, -1]], AppendTo[a, 1], If[varfourier[Append[a, 1]]>varfourier[Append[a, -1]], AppendTo[a, -1], AppendTo[a, -a[[-1]]]]], {j, nmax}];
a=a/.{-1->0};
Print[a]
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Andres Cicuttin, Feb 12 2017
STATUS
approved