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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.
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
Sequence in context: A067551 A080119 A162257 * A366237 A369269 A369300
KEYWORD
nonn,easy
AUTHOR
Roman Witula, Jul 30 2012
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)