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%I #47 Mar 13 2020 08:32:27
%S 3,-4,10,-25,66,-179,493,-1369,3818,-10672,29865,-83626,234237,
%T -656205,1838483,-5151080,14432666,-40438941,113306686,-317477255,
%U 889550021,-2492461633,6983719214,-19567941936,54828148469,-153625048854,430447808073,-1206087937261,3379383275971,-9468821484028
%N 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.
%C a(n) is an integer.
%C This is other half of A215076.
%C 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
%D 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.
%H Colin Barker, <a href="/A274220/b274220.txt">Table of n, a(n) for n = 0..1000</a>
%H B. C. Berndt, L.-C. Zhang, <a href="http://dx.doi.org/10.1007/BF01444636">Ramanujan's identities for eta-functions</a>, Math. Ann. 292 (1992), 561-573.
%H Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Witula/witula17.html">Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7</a>, J. Integer Seq., 12 (2009), Article 09.8.5.
%H Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Witula/witula30.html">Full Description of Ramanujan Cubic Polynomials</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
%H Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Witula2/witula40r.html">Ramanujan Cubic Polynomials of the Second Kind</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
%H Roman Witula, <a href="https://doi.org/10.1515/dema-2013-0418">Ramanujan Type Trigonometric Formulae</a>, Demonstratio Math. 45 (2012) 779-796.
%H Roman Witula and Damian Slota, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota/witula13.html">New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6.
%H Roman Witula, Damian Slota and Adam Warzynski, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Slota/slota57.html">Quasi-Fibonacci Numbers of the Seventh Order</a>, J. Integer Seq., 9 (2006), Article 06.4.3.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-4,-3,1).
%F a(n) = -4*a(n-1)-3*a(n-2)+a(n-3).
%F G.f.: (3+8*x+3*x^2) / (1+4*x+3*x^2-x^3). - _Colin Barker_, Jun 14 2016
%e a(0) = 3, a(1) = -4, a(2) = 10, a(3) = -25.
%t 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 *)
%o (PARI) Vec((3+8*x+3*x^2)/(1+4*x+3*x^2-x^3) + O(x^30)) \\ _Colin Barker_, Jun 14 2016
%o (PARI) polsym(x^3 + 4*x^2 + 3*x - 1,33) \\ _Joerg Arndt_, Mar 12 2020
%Y Cf. A215076, A033304, A094648, A274032.
%K sign,easy
%O 0,1
%A _Kai Wang_, Jun 14 2016
%E Many terms corrected by _Colin Barker_, Jun 14 2016
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