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%I #15 Feb 15 2024 04:42:57
%S 0,-1,21,85,365,-5707,-24935,-108872,1713705,7480420,32652893,
%T -513913649,-2243303605,-9792325686,154118686736,672748988550,
%U 2936640671285,-46218967738367,-201752069488280,-880675175822422,13860700755359325,60503840705600655,264107479466296733
%N The Berndt-type sequence number 8 for the argument 2*Pi/13.
%C a(n) is defined by the relation A217548(n) + a(n)*sqrt(13)= 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, A(1) = sqrt((13-3*sqrt(13))/26).
%C However 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 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.
%e We have A(1) = A(-1) = sqrt((13-3*sqrt(13))/2), A(2) = (7-sqrt(13))/2, A(3) = (2*sqrt(13)-3)*sqrt((13-3*sqrt(13))/26), A(4) = (21-5*sqrt(13))/2, A(5) = ((13*sqrt(13)-37)/2)*sqrt((13-3*sqrt(13))/26), 2*sqrt(13)*A(6) = -295 + 85*sqrt(13), and 2*sqrt(13)*(A(6) - 4*A(4)) + 2*A(2) = -28. Furthermore it can be verified that -a(5)/13 - a(4) - a(3) = A217548(5)/13 + A217548(4) + A217548(3) = -11.
%Y Cf. A216605, A216486, A216597, A216508, A216540, A161905, A217548, A216450, A216801, A216861.
%K sign
%O 0,3
%A _Roman Witula_, Oct 06 2012
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