A274664 Sum of n-th powers of the roots of x^3 + 11*x^2 - 4*x - 1.

3, -11, 129, -1460, 16565, -187926, 2131986, -24186985, 274396853, -3112981337, 35316195134, -400655674969, 4545364223858, -51566312967180, 585010243859443, -6636832570098735, 75293632933556677, -854192282305658944, 9690652804526376357, -109938656346079219026, 1247233638742671255770

Offset

0,1

Comments

This is the other half for A274663.

a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial x^3 + 11*x^2 - 3*x - 1.

x1 = (cos(2*Pi/7)*cos(4*Pi/7))/(cos(Pi/7))^2,

x2 = -(cos(4*Pi/7)*cos(Pi/7))/(cos(2*Pi/7))^2,

x3 = -(cos(Pi/7) *cos(2*Pi/7))(cos(4*Pi/7))^2.

Links

Colin Barker, Table of n, a(n) for n = 0..900 [corrected by Paolo Xausa, Feb 03 2026]

Index entries for linear recurrences with constant coefficients, signature (-11,4,1).

Formula

a(0) = 3, a(1) = -11, a(2) = 129; thereafter a(n) = -11*a(n-1) + 4*a(n-2) + a(n-3).

a(n) = ((cos(2*Pi/7)*cos(4*Pi/7))/(cos(Pi/7))^2)^n +(-(cos(4*Pi/7)*cos(Pi/7))/(cos(2*Pi/7))^2)^n +(-(cos(Pi/7)*cos(2*Pi/7))/(cos(4*Pi/7))^2)^n.

G.f.: (3+22*x-4*x^2+149090*x^4+1639990*x^5-596360*x^6-149090*x^7) / (1+11*x-4*x^2-x^3). - Colin Barker, Jul 03 2016

Mathematica

A274664[n_] := RootSum[#^3 + 11*#^2 - 4*# - 1 &, #^n &]; Array[A274664, 25, 0] (* or *)
LinearRecurrence[{-11, 4, 1}, {3, -11, 129}, 25] (* Paolo Xausa, Feb 03 2026 *)

Prog

(PARI) Vec((3+22*x-4*x^2+149090*x^4+1639990*x^5-596360*x^6-149090*x^7) / (1+11*x-4*x^2-x^3) + O(x^20)) \\ Colin Barker, Jul 03 2016
(PARI) first(n)=my(x='x); polsym(x^3+11*x^2-4*x-1, n) \\ Charles R Greathouse IV, Jul 10 2016

Crossrefs

Cf. A274663.

Sequence in context: A113258 A113848 A287429 * A219620 A382199 A201611

Adjacent sequences: A274661 A274662 A274663 * A274665 A274666 A274667

Keyword

sign,easy

Author

Kai Wang, Jul 01 2016

Extensions

a(4) corrected by Paolo Xausa, Feb 03 2026

Status

approved