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).

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(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.

All terms are even.

Links

Table of n, a(n) for n=0..15.

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 (78,-715,1716,-1287,286,-13).

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

Adjacent sequences: A353408 A353409 A353410 * A353412 A353413 A353414

Keyword

nonn,easy,changed

Author

Bernard Schott, Apr 19 2022

Status

approved