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 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). 4
 4, 36, 1044, 33300, 1070244, 34420356, 1107069876, 35607151476, 1145248326468, 36835122753252, 1184744167077204, 38105444942929620, 1225602095970073572, 39419576386043222340, 1267869080483029127412, 40779027899804602385460, 1311593714249667915837060, 42185362424185765127267748 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Sum_ {k=1..(m-1)/2)} tan^(2n) (k*Pi/m) is an integer when m >= 3 is an odd integer (see AMM link); this sequence is for the case m = 9. Note tan(3*Pi/9) = tan(Pi/3) = sqrt(3). LINKS Table of n, a(n) for n=0..17. Michel Bataille and Li Zhou, A Combinatorial Sum Goes on Tangent, The American Mathematical Monthly, Vol. 112, No. 7 (Aug. - Sep., 2005), Problem 11044, pp. 657-659. Index entries for linear recurrences with constant coefficients, signature (36,-126,84,-9). FORMULA 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 a(n) = A215948(n) + 3^n. - Jianing Song, Apr 19 2022 EXAMPLE a(1) = tan^2 (Pi/9) + tan^2 (2*Pi/9) + tan^2 (3*Pi/9) + tan^2 (4*Pi/9) = 36. MATHEMATICA LinearRecurrence[{36, -126, 84, -9}, {4, 36, 1044, 33300}, 18] (* Amiram Eldar, Apr 18 2022 *) CROSSREFS Similar with: A000244 (m=3), 2*A165225 (m=5), A108716 (m=7), this sequence (m=9), A275546 (m=11), A353411 (m=13). Cf. A019676 (Pi/9), A019918 (tan(Pi/9)), A019938 (tan(2*Pi/9)). Cf. A215948. Sequence in context: A136469 A120605 A337851 * A360725 A173212 A143764 Adjacent sequences: A353407 A353408 A353409 * A353411 A353412 A353413 KEYWORD nonn,easy AUTHOR Bernard Schott, Apr 17 2022 EXTENSIONS More terms from Stefano Spezia, Apr 18 2022 STATUS approved

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Last modified November 30 03:37 EST 2023. Contains 367452 sequences. (Running on oeis4.)