The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A274075 Sum of n-th powers of the roots of x^3 + x^2 - 9*x - 1. 7
 3, -1, 19, -25, 195, -401, 2131, -5545, 24323, -72097, 285459, -910009, 3407043, -11311665, 41065043, -139462985, 497736707, -1711838529, 6052005907, -20960815961, 73717030595, -256312368337, 898804827731, -3131899112169, 10964830193411, -38253117375201 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS a(n) is always an integer. This is the other half of A274032. a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial x^3 + x^2 - 9*x - 1. x1 = tan(Pi/7)/tan(4*Pi/7), x2 = tan(4*Pi/7)/tan(2*Pi/7), x3 = tan(2*Pi/7)/tan(Pi/7). LINKS Colin Barker, 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 (-1,9,1). FORMULA a(n) = (tan(Pi/7)/tan(4*Pi/7))^n + (tan(4*Pi/7)/tan(2*Pi/7))^n + (tan(2*Pi/7)/tan(Pi/7))^n. a(n) = -a(n-1) + 9*a(n-2) + a(n-3) for n>2. G.f.: (3+2*x-9*x^2) / (1+x-9*x^2-x^3). - Colin Barker, Jun 11 2016 MATHEMATICA FullSimplify[Table[(Tan[Pi/7]/Tan[4*Pi/7])^n + (Tan[4*Pi/7]/Tan[2*Pi/7])^n + (Tan[2*Pi/7]/Tan[Pi/7])^n, {n, 0, 12}]] (* Wesley Ivan Hurt, Jun 11 2016 *) PROG (PARI) Vec((3+2*x-9*x^2)/(1+x-9*x^2-x^3) + O(x^30)) \\ Colin Barker, Jun 11 2016 (PARI) polsym(x^3 + x^2 - 9*x - 1, 30) \\ Charles R Greathouse IV, Jul 20 2016 CROSSREFS Cf. A033304, A094648, A215076, A274032. Sequence in context: A027537 A192721 A002380 * A038455 A343890 A067802 Adjacent sequences:  A274072 A274073 A274074 * A274076 A274077 A274078 KEYWORD sign,easy AUTHOR Kai Wang, Jun 09 2016 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 10 04:27 EDT 2022. Contains 356029 sequences. (Running on oeis4.)