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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. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
a(n) is an integer.
This is other half of A215076.
a(n) is the sum of n-th powers of the roots of x^3 + 4*x^2 + 3*x - 1. - Greg Dresden, Mar 11 2020
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
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.
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
EXAMPLE
a(0) = 3, a(1) = -4, a(2) = 10, a(3) = -25.
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 *)
PROG
(PARI) Vec((3+8*x+3*x^2)/(1+4*x+3*x^2-x^3) + O(x^30)) \\ Colin Barker, Jun 14 2016
(PARI) polsym(x^3 + 4*x^2 + 3*x - 1, 33) \\ Joerg Arndt, Mar 12 2020
CROSSREFS
Sequence in context: A143108 A169790 A014009 * A299881 A218293 A288110
KEYWORD
sign,easy
AUTHOR
Kai Wang, Jun 14 2016
EXTENSIONS
Many terms corrected by Colin Barker, Jun 14 2016
STATUS
approved

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Last modified March 29 05:16 EDT 2024. Contains 371264 sequences. (Running on oeis4.)