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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. 6
 0, 2, 7, 33, 143, 634, 2793, 12326, 54370, 239859, 1058123, 4667893, 20592276, 90842309, 400748476, 1767891558, 7799007839, 34405121341, 151777302615, 669561643730, 2953753868221, 13030408769658, 57483311162030, 253586139972259, 1118688695658615 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. 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. (in review) LINKS Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796. Index entries for linear recurrences with constant coefficients, signature (3,6,1). FORMULA G.f.: (2*x+x^2)/(1-3*x-6*x^2-x^3). MATHEMATICA LinearRecurrence[{3, 6, 1}, {0, 2, 7}, 40] (* T. D. Noe, Jul 30 2012 *) CoefficientList[Series[(2x+x^2)/(1-3x-6x^2-x^3), {x, 0, 30}], x] (* Harvey P. Dale, Sep 13 2021 *) PROG (PARI) Vec((2*x+x^2)/(1-3*x-6*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012 CROSSREFS Cf. A214779, A214778, A214951, A214699. Sequence in context: A067551 A080119 A162257 * A055724 A301433 A054727 Adjacent sequences:  A214951 A214952 A214953 * A214955 A214956 A214957 KEYWORD nonn,easy AUTHOR Roman Witula, Jul 30 2012 STATUS approved

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Last modified July 3 16:27 EDT 2022. Contains 355055 sequences. (Running on oeis4.)