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A353411
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).
4
6, 78, 4654, 312390, 21167510, 1435594238, 97371674686, 6604463476598, 447963730184230, 30384227802426030, 2060884053792801614, 139784466963241906598, 9481221017869954060214, 643086846082033986242142, 43618927438218948551328606, 2958559706907951258983758550
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 the particular case m = 13.
All terms are even.
LINKS
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.
FORMULA
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
EXAMPLE
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.
MATHEMATICA
LinearRecurrence[{78, -715, 1716, -1287, 286, -13}, {6, 78, 4654, 312390, 21167510, 1435594238}, 16] (* Amiram Eldar, Apr 19 2022 *)
CROSSREFS
Similar to: A000244 (m=3), 2*A165225 (m=5), A108716 (m=7), A353410 (m=9), A275546 (m=11), this sequence (m=13).
Sequence in context: A219435 A219135 A208308 * A053771 A294992 A093033
KEYWORD
nonn,easy
AUTHOR
Bernard Schott, Apr 19 2022
STATUS
approved