Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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