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
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
KEYWORD
nonn,easy
AUTHOR
Bernard Schott, Apr 17 2022
EXTENSIONS
More terms from Stefano Spezia, Apr 18 2022
STATUS
approved