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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.
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%I #79 Jan 15 2016 16:31:38

%S 0,1,1,2,2,3,7,9,11,27,34,43,104,131,165,400,504,635,1539,1939,2443,

%T 5921,7460,9399,22780,28701,36161,87642,110422,139123,337187,424829,

%U 535251,1297267,1634454,2059283,4991004,6288271,7922725,19202000,24193004,30481275,73876279,93078279,117271283

%N 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.

%C These three sequences:

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

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

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

%C 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.

%C 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).

%H Alexander Samokrutov, <a href="/A237988/b237988.txt">Table of n, a(n) for n = 0..83</a>

%H Alexander Samokrutov, <a href="/A237988/a237988_1.txt">The roots of the equation of the third degree</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 3, 0, 0, 3, 0, 0, 1).

%F 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

%t 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 *)

%t 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 *)

%o (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

%Y Cf. A247344, A255983, A255985.

%K nonn,easy

%O 0,4

%A _Alexander Samokrutov_, May 01 2015