%I #33 Feb 15 2024 08:45:47
%S 0,-6,-37,676,2882,12502,-196209,-856850,-3740697,58876883,257003504,
%T 1121852777,-17656510365,-77073076671,-336434457597,5295048110651,
%U 23113603862267,100894018986142,-1587942800101489,-6931585922526870,-30257313674299627,476211413709501353
%N The Berndt-type sequence number 9 for the argument 2*Pi/13.
%C a(n) + A218655(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 follows 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 We note that s(1) + s(3) + s(9) = s(1)^(-1) + s(3)^(-1) + s(9)^(-1) = sqrt((13-3*sqrt(13))/2), sqrt(2*sqrt(13))*(s(1)^(-3) + s(3)^(-3) + s(9)^(-3)) = sqrt((97*sqrt(13)-339), and s(1)^(-9) + s(3)^(-9) + s(9)^(-9) = (131/13)*sqrt(2834 - 786*sqrt(13)).
%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, 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).
%H 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 Cf. A216605, A216486, A216508, A216597, A216540, A161905, A217548, A217549, A216450.
%K sign
%O 0,2
%A _Roman Witula_, Oct 25 2012