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%I #34 Dec 26 2024 20:29:32
%S 6,78,4654,312390,21167510,1435594238,97371674686,6604463476598,
%T 447963730184230,30384227802426030,2060884053792801614,
%U 139784466963241906598,9481221017869954060214,643086846082033986242142,43618927438218948551328606,2958559706907951258983758550
%N a(n) = (tan(1*Pi/13))^(2*n) + (tan(2*Pi/13))^(2*n) + (tan(3*Pi/13))^(2*n) + (tan(4*Pi/13))^(2*n) + (tan(5*Pi/13))^(2*n) + (tan(6*Pi/13))^(2*n).
%C Sum_{k=1..(m-1)/2} (tan(k*Pi/m))^(2*n) is an integer when m >= 3 is an odd integer (see AMM link); this sequence is the particular case m = 13.
%C All terms are even.
%H Michel Bataille and Li Zhou, <a href="https://doi.org/10.2307/30037561">A Combinatorial Sum Goes on Tangent</a>, The American Mathematical Monthly, Vol. 112, No. 7 (Aug. - Sep., 2005), Problem 11044, pp. 657-659.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (78,-715,1716,-1287,286,-13).
%F G.f.: -2*(143*x^5 -1287*x^4 +2574*x^3 -1430*x^2 +195*x -3) / (13*x^6 -286*x^5 +1287*x^4 -1716*x^3 +715*x^2 -78*x +1). - _Alois P. Heinz_, Apr 19 2022
%e a(1) = tan^2 (Pi/13) + tan^2 (2*Pi/13) + tan^2 (3*Pi/13) + tan^2 (4*Pi/13) + tan^2 (5*Pi/13) + tan^2 (6*Pi/13) = 78.
%t LinearRecurrence[{78, -715, 1716, -1287, 286, -13}, {6, 78, 4654, 312390, 21167510, 1435594238}, 16] (* _Amiram Eldar_, Apr 19 2022 *)
%Y Similar to: A000244 (m=3), 2*A165225 (m=5), A108716 (m=7), A353410 (m=9), A275546 (m=11), this sequence (m=13).
%K nonn,easy,changed
%O 0,1
%A _Bernard Schott_, Apr 19 2022