%I #26 Jul 20 2016 23:51:54
%S 3,-1,19,-25,195,-401,2131,-5545,24323,-72097,285459,-910009,3407043,
%T -11311665,41065043,-139462985,497736707,-1711838529,6052005907,
%U -20960815961,73717030595,-256312368337,898804827731,-3131899112169,10964830193411,-38253117375201
%N Sum of n-th powers of the roots of x^3 + x^2 - 9*x - 1.
%C a(n) is always an integer.
%C This is the other half of A274032.
%C a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial
%C x^3 + x^2 - 9*x - 1.
%C x1 = tan(Pi/7)/tan(4*Pi/7),
%C x2 = tan(4*Pi/7)/tan(2*Pi/7),
%C x3 = tan(2*Pi/7)/tan(Pi/7).
%H Colin Barker, <a href="/A274075/b274075.txt">Table of n, a(n) for n = 0..1000</a>
%H B. C. Berndt, L.-C. Zhang, <a href="http://dx.doi.org/10.1007/BF01444636">Ramanujan's identities for eta-functions</a>, Math. Ann. 292 (1992), 561-573.
%H Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Witula/witula17.html">Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7</a>, J. Integer Seq., 12 (2009), Article 09.8.5.
%H Roman Witula and Damian Slota, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota/witula13.html">New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
%H Roman Witula and Damian Slota, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota2/slota99.html">Quasi-Fibonacci Numbers of Order 11</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.5
%H Roman Witula, Damian Slota and Adam Warzynski, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Slota/slota57.html">Quasi-Fibonacci Numbers of the Seventh Order</a>, J. Integer Seq., 9 (2006), Article 06.4.3.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,9,1).
%F 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.
%F a(n) = -a(n-1) + 9*a(n-2) + a(n-3) for n>2.
%F G.f.: (3+2*x-9*x^2) / (1+x-9*x^2-x^3). - _Colin Barker_, Jun 11 2016
%t 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 *)
%o (PARI) Vec((3+2*x-9*x^2)/(1+x-9*x^2-x^3) + O(x^30)) \\ _Colin Barker_, Jun 11 2016
%o (PARI) polsym(x^3 + x^2 - 9*x - 1, 30) \\ _Charles R Greathouse IV_, Jul 20 2016
%Y Cf. A033304, A094648, A215076, A274032.
%K sign,easy
%O 0,1
%A _Kai Wang_, Jun 09 2016