A274032 Sum of n-th powers of the roots of x^3 + 9*x^2 - x - 1.

3, -9, 83, -753, 6851, -62329, 567059, -5159009, 46935811, -427014249, 3884905043, -35344223825, 321555905219, -2925462465753, 26615373873171, -242142271419073, 2202970354179075, -20042260085157577, 182341168849178195, -1658909809373582257

Offset: 0

Author

Kai Wang, Jun 07 2016

Keywords

• sign
• easy

Comments

A Berndt-type sequence for tan(2*Pi/7).

a(n) is always an integer.

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

x^3 + 9*x^2 - x - 1.

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

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

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

This is a two sided sequence. The other half is A274075. - Kai Wang, Aug 02 2016

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• B. C. Berndt, L.-C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573.
• Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.
• Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
• Roman Witula and Damian Slota, Quasi-Fibonacci Numbers of Order 11, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.5
• Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
• Index entries for linear recurrences with constant coefficients, signature (-9,1,1).

Crossrefs

Cf. A033304, A094648, A215076, A274075.

Programs

PARI

Vec((3+18*x-x^2)/(1+9*x-x^2-x^3) + O(x^30)) \\ Colin Barker, Jun 07 2016

PARI

polsym(x^3 + 9*x^2 - x - 1, 30) \\ Charles R Greathouse IV, Jul 20 2016

Formula

a(n) = (tan(Pi/7)/tan(2*Pi/7))^n + (-tan(2*Pi/7)/tan(3*Pi/7))^n + (-tan(3*Pi/7)/tan(Pi/7))^n.

From Colin Barker, Jun 07 2016: (Start)

a(n) = -9*a(n-1)+a(n-2)+a(n-3) for n>2.

G.f.: (3+18*x-x^2) / (1+9*x-x^2-x^3).

(End)

Extensions

Edited by N. J. A. Sloane, Jun 07 2016