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A214954 a(n) = 3*a(n-1) + 6*a(n-2) + a(n-3), with a(0) = 0, a(1) = 2, and a(2) = 7.

%I A214954 #24 Sep 13 2021 18:25:37
%S A214954 0,2,7,33,143,634,2793,12326,54370,239859,1058123,4667893,20592276,
%T A214954 90842309,400748476,1767891558,7799007839,34405121341,151777302615,
%U A214954 669561643730,2953753868221,13030408769658,57483311162030,253586139972259,1118688695658615
%N A214954 a(n) = 3*a(n-1) + 6*a(n-2) + a(n-3), with a(0) = 0, a(1) = 2, and a(2) = 7.
%C A214954 Ramanujan-type sequence number 5 for the argument 2*Pi/9 is defined by the following relation: 81^(1/3)*a(n)=(c(1)/c(2))^(n + 1/3) + (c(2)/c(4))^(n + 1/3) + (c(4)/c(1))^(n + 1/3), where c(j) := Cos(2Pi*j/9) - for the proof see Witula's et al. papers. We have a(n)=cx(3n+1), where the sequence cx(n) and its two conjugate sequences ax(n) and bx(n) are defined in the comments to the sequence A214779. We note that ax(3n+1)=bx(3n+1)=0. Further we have ax(3n)=A214778(n), bx(3n)=cx(3n)=0 and bx(3n-1)=A214951(n), ax(3n-1)=cx(3n-1)=0.
%D A214954 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. (in review)
%H A214954 Roman Witula, <a href="https://doi.org/10.1515/dema-2013-0418">Ramanujan Type Trigonometric Formulae</a>, Demonstratio Math. 45 (2012) 779-796.
%H A214954 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,1).
%F A214954 G.f.: (2*x+x^2)/(1-3*x-6*x^2-x^3).
%t A214954 LinearRecurrence[{3, 6, 1}, {0, 2, 7}, 40] (* _T. D. Noe_, Jul 30 2012 *)
%t A214954 CoefficientList[Series[(2x+x^2)/(1-3x-6x^2-x^3),{x,0,30}],x] (* _Harvey P. Dale_, Sep 13 2021 *)
%o A214954 (PARI) Vec((2*x+x^2)/(1-3*x-6*x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Oct 01 2012
%Y A214954 Cf. A214779, A214778, A214951, A214699.
%K A214954 nonn,easy
%O A214954 0,2
%A A214954 _Roman Witula_, Jul 30 2012