A274220 a(n) = (-cos(Pi/7)/cos(2*Pi/7))^n + (-cos(2*Pi/7)/cos(3*Pi/7))^n + (cos(3*Pi/7)/cos(Pi/7))^n.
3, -4, 10, -25, 66, -179, 493, -1369, 3818, -10672, 29865, -83626, 234237, -656205, 1838483, -5151080, 14432666, -40438941, 113306686, -317477255, 889550021, -2492461633, 6983719214, -19567941936, 54828148469, -153625048854, 430447808073, -1206087937261, 3379383275971, -9468821484028
Offset: 0
Examples
a(0) = 3, a(1) = -4, a(2) = 10, a(3) = -25.
References
- R. Witula, E. Hetmaniok, D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.
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, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
- Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
- Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
- 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, 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 (-4,-3,1).
Programs
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Mathematica
CoefficientList[Series[(3 + 8 x + 3 x^2)/(1 + 4 x + 3 x^2 - x^3), {x, 0, 29}], x] (* Michael De Vlieger, Jun 14 2016 *) -
PARI
Vec((3+8*x+3*x^2)/(1+4*x+3*x^2-x^3) + O(x^30)) \\ Colin Barker, Jun 14 2016
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PARI
polsym(x^3 + 4*x^2 + 3*x - 1,33) \\ Joerg Arndt, Mar 12 2020
Formula
a(n) = -4*a(n-1)-3*a(n-2)+a(n-3).
G.f.: (3+8*x+3*x^2) / (1+4*x+3*x^2-x^3). - Colin Barker, Jun 14 2016
Extensions
Many terms corrected by Colin Barker, Jun 14 2016
Comments