This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A353410 #36 Dec 26 2024 20:29:24
%S A353410 4,36,1044,33300,1070244,34420356,1107069876,35607151476,
%T A353410 1145248326468,36835122753252,1184744167077204,38105444942929620,
%U A353410 1225602095970073572,39419576386043222340,1267869080483029127412,40779027899804602385460,1311593714249667915837060,42185362424185765127267748
%N A353410 a(n) = (tan(1*Pi/9))^(2*n) + (tan(2*Pi/9))^(2*n) + (tan(3*Pi/9))^(2*n) + (tan(4*Pi/9))^(2*n).
%C A353410 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 for the case m = 9.
%C A353410 Note tan(3*Pi/9) = tan(Pi/3) = sqrt(3).
%H A353410 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 A353410 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (36,-126,84,-9).
%F A353410 G.f.: 4*(1 - 27x + 63*x^2 - 21*x^3)/((1 - 3*x)*(1 - 33*x + 27*x^2 - 3*x^3)). - _Stefano Spezia_, Apr 18 2022
%F A353410 a(n) = A215948(n) + 3^n. - _Jianing Song_, Apr 19 2022
%e A353410 a(1) = tan^2 (Pi/9) + tan^2 (2*Pi/9) + tan^2 (3*Pi/9) + tan^2 (4*Pi/9) = 36.
%t A353410 LinearRecurrence[{36, -126, 84, -9}, {4, 36, 1044, 33300}, 18] (* _Amiram Eldar_, Apr 18 2022 *)
%Y A353410 Similar with: A000244 (m=3), 2*A165225 (m=5), A108716 (m=7), this sequence (m=9), A275546 (m=11), A353411 (m=13).
%Y A353410 Cf. A019676 (Pi/9), A019918 (tan(Pi/9)), A019938 (tan(2*Pi/9)).
%Y A353410 Cf. A215948.
%K A353410 nonn,easy
%O A353410 0,1
%A A353410 _Bernard Schott_, Apr 17 2022
%E A353410 More terms from _Stefano Spezia_, Apr 18 2022