A214778 a(n) = 3*a(n-1) + 6*a(n-2) + a(n-3), with a(0) = 3, a(1) = 3, and a(2) = 21.

3, 3, 21, 84, 381, 1668, 7374, 32511, 143445, 632775, 2791506, 12314613, 54325650, 239656134, 1057236915, 4663973199, 20574997221, 90766067772, 400412159841, 1766407883376, 7792462676946, 34376247490935, 151649926417857, 668999726876127, 2951274986626458

Offset

0,1

Comments

Ramanujan-type sequence number 3 for the argument 2Pi/9 is equal to the subsequence ax(3n) of the sequence ax(n), which (with its two conjugate sequences bx(n) and cx(n)) is defined in the comments to the sequence A214779 (we note that simultaneously we have bx(3n)=cx(3n)=0).

From example below follows that a(n) is equal to the sum of n-th powers of the roots of the polynomial x^3-3*x^2-6*x-1.

We note that all a(n) are divisible by 3 and a(n)/3 == 1 (mod 3). - Roman Witula, Oct 06 2012

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.

Links

Table of n, a(n) for n=0..24.

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

a(n) = (c(1)/c(2))^n + (c(2)/c(4))^n + (c(4)/c(1))^n, where c(j) := Cos(2*Pi*j/9).

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

a(n+1) = A214951(n+1) - A214951(n). - Roman Witula, Oct 06 2012

Example

From a(1)=3 (after squaring) and a(2)=21 the following equality follows c(1)/c(4) + c(4)/c(2) + c(2)/c(1) = -6, which implies the decomposition x^3 - 3*x^2 - 6*x - 1 =(x - c(1)/c(2))*(x - c(2)/c(4))*(x - c(4)/c(1)).

Mathematica

LinearRecurrence[{3, 6, 1}, {3, 3, 21}, 40] (* T. D. Noe, Jul 30 2012 *)

Prog

(PARI) Vec((3-6*x-6*x^2)/(1-3*x -6*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012
(PARI) polsym(x^3 - 3*x^2 - 6*x - 1, 30) \\ Charles R Greathouse IV, Jul 20 2016

Crossrefs

Cf. A214699, A214779.

Sequence in context: A369078 A209528 A369751 * A180754 A224091 A224751

Adjacent sequences: A214775 A214776 A214777 * A214779 A214780 A214781

Keyword

nonn,easy

Author

Roman Witula, Jul 28 2012

Status

approved