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A217548
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The Berndt-type sequences number 7 for the argument 2*Pi/13.
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5
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6, 7, -65, -295, -1303, 20631, 89967, 392616, -6178549, -26970688, -117731275, 1852943703, 8088348131, 35306734632, -555682818080, -2425630962790, -10588208505263, 166644858132571, -727427431532172, 3175319503526856, -49975467287014789
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OFFSET
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0,1
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COMMENTS
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a(n) is the rational component (with respect to the field Q(sqrt(13))) of the number A(2*n)*2*13^(floor((n+1)/3)/2), where A(n) = sqrt((13-3*sqrt(13))/2)*A(n-1) + (sqrt(13)-3)*A(n-2)/2 - sqrt((13-3*sqrt(13))/26)*A(n-3), with A(-1) = sqrt((13-3*sqrt(13))/2), A(0)=3, and A(1) = sqrt((13-3*sqrt(13))/2).
The basic sequence A(n) is defined by the relation A(n) = s(1)^(-n) + s(3)^(-n) + s(9)^(-n), where s(j) = 2*sin(2*Pi*j/13). The sequence with respective positive powers is discussed in A216508 (see sequence Y(n) in Comments to A216508).
We note that s(1) + s(3) + s(9) = s(1)^(-1) + s(3)^(-1) + s(9)^(-1) = sqrt((13-3*sqrt(13))/2).
The numbers of other Berndt-type sequences for the argument 2*Pi/13 in crossrefs are given.
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REFERENCES
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R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, Thirteenth International Conference on Fibonacci Numbers and their Applications, Congressus Numerantium, 201 (2010), 89-107.
R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).
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LINKS
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CROSSREFS
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Cf. A019698, A216605, A216486, A216508, A216597, A216540, A161905, A216450, A216801, A216861, A217548, A217549, A211988.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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