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A218655 The Berndt-type sequence number 10 for the argument 2*Pi/13. 1


%S 2,4,13,-176,-786,-3452,54483,237722,1037569,-16329149,-71279530,

%T -311145495,4897036897,21376227709,93310132523,-1468582101731,

%U -6410560285891,-27982966049682,440416091468393,1922476035761802,8391868916275609

%N The Berndt-type sequence number 10 for the argument 2*Pi/13.

%C A211988(n) + a(n)*sqrt(13) = A(2*n+1)*13^((1 + floor(n/3))/2)*sqrt(2*(13 + 3*sqrt(13))/13), where A(n) is defined below.

%C The sequence A(n) from the name of 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 It could be deduced that 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 The numbers of other Berndt-type sequences for the argument 2*Pi/13 in crossrefs are given.

%D R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13 on the occasion of the Thirteenth International Conference on Fibonacci Numbers and their Applications, Congressus Numerantium, 201 (2010), 89-107.

%D R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

%e Let us put b(n) = A211988(n) + a(n)*sqrt(13). Then we get b(0) = 2*sqrt(13), b(1) = -6 + 4*sqrt(13), b(2) = -37 + 13*sqrt(13), b(3) = 676 - 176*sqrt(13), b(4) = 2882 - 786*sqrt(13), b(5) = 12502 - 3452*sqrt(13).

%Y Cf. A211988, A216605, A216486, A216508, A216597, A216540, A161905, A217548, A217549, A216450.

%K sign

%O 0,1

%A _Roman Witula_, Nov 04 2012

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Last modified July 9 16:50 EDT 2020. Contains 335545 sequences. (Running on oeis4.)