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Sum of n-th powers of the roots of x^3 -31* x^2 - 25*x - 1.
3

%I #44 Feb 02 2022 13:49:53

%S 3,31,1011,32119,1020995,32454831,1031656755,32793751175,

%T 1042430160131,33136210400191,1053316070160371,33482245865136407,

%U 1064315659783638083,33831894915991351119,1075430116136187973171,34185195288781394584359,1086660638750543922795523

%N Sum of n-th powers of the roots of x^3 -31* x^2 - 25*x - 1.

%C This is one side of a two sided sequence (see A248417).

%C a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial

%C x^3 -31* x^2 - 25*x - 1.

%C x1 = (tan(Pi/7))^2/(tan(2*Pi/7)*tan(4*Pi/7)),

%C x2 = (tan(2*Pi/7))^2/(tan(4*Pi/7)*tan(Pi/7)),

%C x3 = (tan(4*Pi/7))^2/(tan(Pi/7)*tan(2*Pi/7)).

%H Colin Barker, <a href="/A274592/b274592.txt">Table of n, a(n) for n = 0..600</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (31,25,1).

%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) = 31*a(n-1) + 25*a(n-2) + a(n-3).

%F G.f.: (3-62*x-25*x^2) / (1-31*x-25*x^2-x^3). - _Colin Barker_, Jun 30 2016

%t LinearRecurrence[{31,25,1},{3,31,1011},20] (* _Harvey P. Dale_, Feb 02 2022 *)

%o (PARI) Vec((3-62*x-25*x^2)/(1-31*x-25*x^2-x^3) + O(x^20)) \\ _Colin Barker_, Jun 30 2016

%o (PARI) polsym(x^3 -31* x^2 - 25*x - 1, 30) \\ _Charles R Greathouse IV_, Jul 20 2016

%Y Cf. A248417, A274032, A274075.

%K nonn,easy

%O 0,1

%A _Kai Wang_, Jun 29 2016