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%I #48 Jul 20 2016 23:51:51
%S 3,-25,563,-13297,314947,-7460905,176745971,-4187046273,99189570819,
%T -2349764090041,55665038509363,-1318684086371985,31239136201419331,
%U -740043533319442377,17531356426655688179,-415311321997288071457,9838570957172556010499,-233072091590971314359129,5521391278779936334581299
%N Sum of n-th powers of the roots of x^3 +25* x^2 + 31*x - 1.
%C This is the other half of A274592.
%C a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial
%C x^3 +25* x^2 + 31*x - 1.
%C x1 = (tan(2*Pi/7)*tan(4*Pi/7))/(tan(Pi/7))^2,
%C x2 = (tan(4*Pi/7)*tan(Pi/7))/(tan(2*Pi/7))^2,
%C x3 = (tan(Pi/7)*tan(2*Pi/7))(tan(4*Pi/7))^2.
%H Colin Barker, <a href="/A248417/b248417.txt">Table of n, a(n) for n = 0..700</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-25,-31,1).
%F a(n) = ((tan(Pi/7))^2/(tan(2*Pi/7)*tan(4*Pi/7)))^(-n)+((tan(2*Pi/7))^2/(tan(4*Pi/7)*tan(Pi/7)))^(-n)+((tan(4*Pi/7))^2/(tan(Pi/7)*tan(2*Pi/7)))^(-n).
%F a(n) = -25*a(n-1) - 31*a(n-2) + a(n-3).
%F G.f.: (3+50*x+31*x^2) / (1+25*x+31*x^2-x^3). - _Colin Barker_, Jul 01 2016
%t CoefficientList[Series[(3 + 50 x + 31 x^2)/(1 + 25 x + 31 x^2 - x^3), {x, 0, 18}], x] (* _Michael De Vlieger_, Jul 01 2016 *)
%o (PARI) Vec((3+50*x+31*x^2)/(1+25*x+31*x^2-x^3) + O(x^20)) \\ _Colin Barker_, Jul 01 2016
%o (PARI) polsym(x^3 +25* x^2 + 31*x - 1, 30) \\ _Charles R Greathouse IV_, Jul 20 2016
%Y Cf. A274592.
%K sign,easy
%O 0,1
%A _Kai Wang_, Jul 01 2016
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