A248417 Sum of n-th powers of the roots of x^3 +25* x^2 + 31*x - 1.

3, -25, 563, -13297, 314947, -7460905, 176745971, -4187046273, 99189570819, -2349764090041, 55665038509363, -1318684086371985, 31239136201419331, -740043533319442377, 17531356426655688179, -415311321997288071457, 9838570957172556010499, -233072091590971314359129, 5521391278779936334581299

Offset

0,1

Comments

This is the other half of A274592.

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

x^3 +25* x^2 + 31*x - 1.

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

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

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

Links

Colin Barker, Table of n, a(n) for n = 0..700

Index entries for linear recurrences with constant coefficients, signature (-25,-31,1).

Formula

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).

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

G.f.: (3+50*x+31*x^2) / (1+25*x+31*x^2-x^3). - Colin Barker, Jul 01 2016

Mathematica

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 *)

Prog

(PARI) Vec((3+50*x+31*x^2)/(1+25*x+31*x^2-x^3) + O(x^20)) \\ Colin Barker, Jul 01 2016
(PARI) polsym(x^3 +25* x^2 + 31*x - 1, 30) \\ Charles R Greathouse IV, Jul 20 2016

Crossrefs

Cf. A274592.

Sequence in context: A224679 A213599 A179473 * A355123 A306795 A131310

Adjacent sequences: A248414 A248415 A248416 * A248418 A248419 A248420

Keyword

sign,easy

Author

Kai Wang, Jul 01 2016

Status

approved