|
%I #27 Oct 08 2016 10:02:12
%S 4,2,10,23,70,197,571,1640,4726,13604,39175,112796,324787,935183,
%T 2692756,7753478,22325254,64283003,185095534,532961345,1534601035,
%U 4418707568,12723161362,36634883048,105485941579,303734663372,874569107071
%N Expansion of (4-6*x-6*x^2+x^3)/((1+x)*(1-3*x+x^3)).
%C Let A_{9,3} = [0,0,0,1; 0,0,1,1; 0,1,1,1; 1,1,1,1], a unit-primitive matrix (see [Jeffery]). Then a(n) = Trace([A_{9,3}]^n).
%H L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2, 3, -1, -1).
%F G.f.: (4-6*x-6*x^2+x^3)/((1+x)*(1-3*x+x^3)).
%F a(n) = 2*a(n-1)+3*a(n-2)-a(n-3)-a(n-4), {a(m)}={4,2,10,23}, m=0,1,2,3.
%F a(n) = Sum_{k=1..4} ((x_k)^3-2*(x_k))^n, x_k=2*(-1)^(k-1)*cos(k*Pi/9).
%F a(n) = (-1)^n+(1+2*cos(Pi/9))^n+(1-cos(Pi/9)+sqrt(3)*sin(Pi/9))^n + (1-cos(Pi/9)-sqrt(3)*sin(Pi/9))^n. - _L. Edson Jeffery_, Dec 15 2011
%F a(n) = (-1)^n + 3*A147704(n). - _R. J. Mathar_, Oct 08 2016
%t CoefficientList[Series[(4-6x-6x^2+x^3)/((1+x)(1-3x+x^3)), {x,0,30}],x] (* or *) LinearRecurrence[{2,3,-1,-1},{4,2,10,23},30] (* _Harvey P. Dale_, Apr 22 2011 *)
%K nonn,easy
%O 0,1
%A _L. Edson Jeffery_, Apr 05 2011
|
