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A216508 a(n) = 13*a(n-1) - 65*a(n-2) + 156*a(n-3) - 182*a(n-4) + 91*a(n-5) - 13*a(n-6), with initial terms 6, 13, 39, 130, 455, 1638.

%I A216508 #52 Feb 08 2025 14:26:50
%S A216508 6,13,39,130,455,1638,6006,22308,83655,316030,1200914,4585308,
%T A216508 17577014,67603887,260757536,1008258225,3906958055,15167837542,
%U A216508 58983478554,229708325847,895760071050,3497141791455,13667427167576,53464307173927,209315686335366,820090746381088,3215215287887889
%N A216508 a(n) = 13*a(n-1) - 65*a(n-2) + 156*a(n-3) - 182*a(n-4) + 91*a(n-5) - 13*a(n-6), with initial terms 6, 13, 39, 130, 455, 1638.
%C A216508 a(n) is equal to the rational part of 2*X(2*n) (with respect to the field Q(sqrt(13))), where X(n) = sqrt((13 + 3*sqrt(13))/2)*X(n-1) - sqrt(13)*X(n-2) + sqrt((13 - 3*sqrt(13))/2)*X(n-3), with X(0)=3, X(1)=sqrt((13 + 3*sqrt(13))/2), and X(2)=(13 - sqrt(13))/2.
%C A216508 The Berndt-type sequence number 3 for the argument 2Pi/13 defined by the relation a(n) + A216597(n)*sqrt(13) = 2*X(2*n), where X(n) := s(2)^n + s(5)^n + s(6)^n, where s(j) := 2*sin(2*Pi*j/13).
%C A216508 We note that all numbers of the form a(6*n+k)*13^(-n), where k = 1,...,6, n = 0,1,... are integers, and even the number a(13)*13^(-4) is an integer.
%C A216508 a(n) is also equal to the rational part of 2*Y(2*n) (with respect to the field Q(sqrt(13))), where Y(n) = sqrt((13 - 3*sqrt(13))/2)*Y(n-1) + sqrt(13)*Y(n-2) - sqrt((13 + 3*sqrt(13))/2)*Y(n-3), with Y(0)=3, Y(1)=sqrt((13 - 3*sqrt(13))/2), and Y(2)=(13 + sqrt(13))/2. Moreover we can deduce the following decompositions:
%C A216508   2*Y(2*n) = a(n) - A216597(n)*sqrt(13) and Y(n) = s(1)^n + s(3)^n + s(9)^n (we have s(9) = -s(4)) - Roman Witula, Sep 22 2012
%D A216508 Roman 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 A216508 Roman Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).
%H A216508 Roman 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.
%H A216508 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (13,-65,156,-182,91,-13).
%F A216508 G.f.: -(91*x^5-364*x^4+468*x^3-260*x^2+65*x-6) / (13*x^6-91*x^5+182*x^4-156*x^3+65*x^2-13*x+1). - _Colin Barker_, Jun 01 2013
%e A216508 We have a(7)/2 + 2*A216597(7) = 26, 4*X(4) - X(6) = 13 + sqrt(13), 4*X(8) - X(10) = 91, 4*X(10) - X(12) = 13*(21 - sqrt(13)), 4*X(12) - X(14)= 78*(11 - sqrt(13)), 8*X(14) - 2*X(16) = 11*13*sqrt(13)*(3*sqrt(13) - 5) and X(6) - 10*X(2) = -6*sqrt(13) since 2*X(2) = 13 - sqrt(13), 2*X(4) = 39 - 5*sqrt(13), X(6) = 65 - 11*sqrt(13), 2*X(8) = 91*(5 - sqrt(13)), X(10) = 91*(9 - 2*sqrt(13)), X(12) = 3003 - 715*sqrt(13) = 13*(3*77 - 55*sqrt(13)), X(14) = 11154 - 2782*sqrt(13), 2*X(16) = 83655 - 21541*sqrt(13).
%t A216508 LinearRecurrence[{13,-65,156,-182,91,-13}, {6,13,39,130,455,1638}, 30]
%Y A216508 Cf. A216605, A216486, A216597, A216540, A161905.
%K A216508 nonn,easy
%O A216508 0,1
%A A216508 _Roman Witula_, Sep 11 2012
%E A216508 Better name from _Joerg Arndt_, Sep 17 2012
%E A216508 Name clarified by _Robert C. Lyons_, Feb 08 2025