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.
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
KEYWORD
nonn,easy
AUTHOR
Bernard Schott, Apr 19 2022
STATUS
approved