A274663 Sum of n-th powers of the roots of x^3 + 4*x^2 - 11*x - 1.

3, -4, 38, -193, 1186, -6829, 40169, -234609, 1373466, -8034394, 47011093, -275049240, 1609284589, -9415668903, 55089756851, -322322100748, 1885860059450, -11033893589177, 64557712909910, -377717821061137, 2209972232664381, -12930227249420121

Offset

0,1

Comments

This is half of a two sided sequences.

The other half is A274664. - Kai Wang, Aug 02 2016

a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial x^3 + 4*x^2 - 11*x - 1.

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

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

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

Links

Colin Barker, Table of n, a(n) for n = 0..1000

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

Formula

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

a(n) = -4*a(n-1) + 11*a(n-2) + a(n-3) for n>2.

G.f.: (3+8*x-11*x^2)/(1+4*x-11*x^2-x^3). - Wesley Ivan Hurt, Jul 02 2016

a(n) = (-1/8)^(-n)*cos(Pi/7)^(3*n) + (-8)^n*sin(Pi/14)^(3*n) +

8^n*sin(3*Pi/14)^(3*n). - Wesley Ivan Hurt, Jul 11 2016

Mathematica

RecurrenceTable[{a[0] == 3, a[1] == -4, a[2] == 38, a[n] == -4 a[n - 1] + 11 a[n - 2] + a[n - 3]}, a, {n, 0, 20}] (* Michael De Vlieger, Jul 02 2016 *)
LinearRecurrence[{-4, 11, 1}, {3, -4, 38}, 30] (* Harvey P. Dale, Dec 28 2022 *)

Prog

(PARI) polsym(x^3 + 4*x^2 - 11*x - 1, 21)
(PARI) Vec((3+8*x-11*x^2)/(1+4*x-11*x^2-x^3) + O(x^99)) \\ Altug Alkan, Jul 08 2016

Crossrefs

Cf. A215076, A248417, A274220, A274592, A274664.

Sequence in context: A102920 A064335 A013330 * A088168 A032836 A167760

Adjacent sequences: A274660 A274661 A274662 * A274664 A274665 A274666

Keyword

sign,easy

Author

Kai Wang, Jul 01 2016

Status

approved