A237988 a(n) = 3*a(n-3) + 3*a(n-6) + a(n-9) for n>8, a(0)=0, a(1)=a(2)=1, a(3)=a(4)=2, a(5)=3, a(6)=7, a(7)=9, a(8)=11.

0, 1, 1, 2, 2, 3, 7, 9, 11, 27, 34, 43, 104, 131, 165, 400, 504, 635, 1539, 1939, 2443, 5921, 7460, 9399, 22780, 28701, 36161, 87642, 110422, 139123, 337187, 424829, 535251, 1297267, 1634454, 2059283, 4991004, 6288271, 7922725, 19202000, 24193004, 30481275, 73876279, 93078279, 117271283

Offset

0,4

Comments

These three sequences:

b(3n+3) = b(3n) + b(3n+1) + b(3n+2),

b(3n+4) = 2*b(3n) + b(3n+1) + b(3n+2),

b(3n+5) = 2*b(3n) + 2*b(3n+1) + b(3n+2),

give the polynomial x^3-3*x^2-3*x-1 with root 1 + 2^(1/3) + 2^(2/3). More generally, see link the roots of the equation of the third degree.

Equation: 4*x^3 - 6*x*y*z + 2*y^3 + z^3 = 3, if x = a(3n), y = a(3n+1), z = a(3n+2).

Links

Alexander Samokrutov, Table of n, a(n) for n = 0..83

Alexander Samokrutov, The roots of the equation of the third degree

Index entries for linear recurrences with constant coefficients, signature (0, 0, 3, 0, 0, 3, 0, 0, 1).

Formula

G.f.: x*(x^7-x^5+x^3-2*x^2-x-1) / (x^9+3*x^6+3*x^3-1). - Colin Barker, May 01 2015

Mathematica

LinearRecurrence[{0, 0, 3, 0, 0, 3, 0, 0, 1}, {0, 1, 1, 2, 2, 3, 7, 9, 11}, 60] (* Vincenzo Librandi, May 15 2015 *)
CoefficientList[ Series[(x^8 - x^6 + x^4 - 2x^3 - x^2 - x)/(x^9 + 3x^6 + 3x^3 - 1), {x, 0, 44}], x] (* Robert G. Wilson v, Jul 24 2015 *)

Prog

(PARI) concat(0, Vec(x*(x^7-x^5+x^3-2*x^2-x-1)/(x^9+3*x^6+3*x^3-1) + O(x^100))) \\ Colin Barker, May 01 2015

Crossrefs

Cf. A247344, A255983, A255985.

Sequence in context: A211395 A160433 A043550 * A291102 A208494 A036060

Adjacent sequences: A237985 A237986 A237987 * A237989 A237990 A237991

Keyword

nonn,easy

Author

Alexander Samokrutov, May 01 2015

Status

approved