OFFSET
0,1
COMMENTS
a(n) is always an integer.
This is the other half of A274032.
a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial
x^3 + x^2 - 9*x - 1.
x1 = tan(Pi/7)/tan(4*Pi/7),
x2 = tan(4*Pi/7)/tan(2*Pi/7),
x3 = tan(2*Pi/7)/tan(Pi/7).
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
B. C. Berndt, L.-C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573.
Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
Roman Witula and Damian Slota, Quasi-Fibonacci Numbers of Order 11, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.5
Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (-1,9,1).
FORMULA
a(n) = (tan(Pi/7)/tan(4*Pi/7))^n + (tan(4*Pi/7)/tan(2*Pi/7))^n + (tan(2*Pi/7)/tan(Pi/7))^n.
a(n) = -a(n-1) + 9*a(n-2) + a(n-3) for n>2.
G.f.: (3+2*x-9*x^2) / (1+x-9*x^2-x^3). - Colin Barker, Jun 11 2016
MATHEMATICA
FullSimplify[Table[(Tan[Pi/7]/Tan[4*Pi/7])^n + (Tan[4*Pi/7]/Tan[2*Pi/7])^n + (Tan[2*Pi/7]/Tan[Pi/7])^n, {n, 0, 12}]] (* Wesley Ivan Hurt, Jun 11 2016 *)
PROG
(PARI) Vec((3+2*x-9*x^2)/(1+x-9*x^2-x^3) + O(x^30)) \\ Colin Barker, Jun 11 2016
(PARI) polsym(x^3 + x^2 - 9*x - 1, 30) \\ Charles R Greathouse IV, Jul 20 2016
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Kai Wang, Jun 09 2016
STATUS
approved