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A217548 The Berndt-type sequences number 7 for the argument 2*Pi/13.

%I A217548 #26 Feb 15 2024 08:45:28
%S A217548 6,7,-65,-295,-1303,20631,89967,392616,-6178549,-26970688,-117731275,
%T A217548 1852943703,8088348131,35306734632,-555682818080,-2425630962790,
%U A217548 -10588208505263,166644858132571,-727427431532172,3175319503526856,-49975467287014789
%N A217548 The Berndt-type sequences number 7 for the argument 2*Pi/13.
%C A217548 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).
%C A217548 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).
%C A217548 We note that s(1) + s(3) + s(9) = s(1)^(-1) + s(3)^(-1) + s(9)^(-1) = sqrt((13-3*sqrt(13))/2).
%C A217548 The numbers of other Berndt-type sequences for the argument 2*Pi/13 in crossrefs are given.
%D A217548 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.
%D A217548 R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).
%H A217548 R. Witula and D. Slota, <a href="https://www.mathstat.dal.ca/fibonacci/abstracts.pdf">Quasi-Fibonacci numbers of order 13</a>, (abstract) see p. 15.
%Y A217548 Cf. A019698, A216605, A216486, A216508, A216597, A216540, A161905, A216450, A216801, A216861, A217548, A217549, A211988.
%K A217548 sign
%O A217548 0,1
%A A217548 _Roman Witula_, Oct 06 2012