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.

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

Offset: 0

Author

Roman Witula, Jul 30 2012

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A214779, A214778, A214951, A214699.

Programs

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 *)

PARI

Vec((2*x+x^2)/(1-3*x-6*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012

Formula

G.f.: (2*x+x^2)/(1-3*x-6*x^2-x^3).