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A077834 Expansion of 1/(1 - 2*x - 2*x^2 - 3*x^3).

%I A077834 #23 Dec 23 2016 21:22:57
%S A077834 1,2,6,19,56,168,505,1514,4542,13627,40880,122640,367921,1103762,
%T A077834 3311286,9933859,29801576,89404728,268214185,804642554,2413927662,
%U A077834 7241782987,21725348960,65176046880,195528140641,586584421922,1759753265766,5279259797299,15837779391896
%N A077834 Expansion of 1/(1 - 2*x - 2*x^2 - 3*x^3).
%H A077834 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,3)
%F A077834 From _Paul Barry_, May 19 2004: (Start)
%F A077834 Convolution of A000244 and A049347.
%F A077834 G.f.: 1/((1-3*x)(1 + x + x^2)).
%F A077834 a(n) = sum_{k=0..n} (3^k*2*sqrt(3)*cos(2*Pi*(n-k)/3 + Pi/6)/3).
%F A077834 a(n) = 3^(n+2)/13 + 2*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/39 + 2*sqrt(3)*sin(2*Pi*n/3 + Pi/3)/13.
%F A077834 (End)
%F A077834 a(n) = A152733(n+3)/3. - _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008
%F A077834 a(0)=1, a(1)=2, a(2)=6, a(n) = 2*a(n-1) + 2*a(n-2) + 3*a(n-3). - _Harvey P. Dale_, Jan 31 2012
%F A077834 a(n) = 1/52*(4*3^(n + 2) + (-1)^n*(2*(-1)^floor(n/3) + 9*(-1)^floor((1 + n)/3) + 6*(-1)^floor((n + 2)/3) + (-1)^floor((n + 4)/3))). - _John M. Campbell_, Dec 23 2016
%p A077834 A049347 := proc(n) op(1+(n mod 3),[1,-1,0]) ; end proc:
%p A077834 A077834 := proc(n) (3^(n+2)+3*A049347(n-1)+4*A049347(n))/13 ; end proc:
%p A077834 seq(A077834(n),n=0..20) ; # _R. J. Mathar_, Mar 22 2011
%t A077834 k0=k1=0;lst={};Do[kt=k1;k1=3^n-k1-k0;k0=kt;AppendTo[lst, k1/3], {n, 1, 5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008 *)
%t A077834 CoefficientList[Series[1/(1-2x-2x^2-3x^3),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,3},{1,2,6},30] (* _Harvey P. Dale_, Jan 31 2012 *)
%o A077834 (PARI) Vec(1/(1-2*x-2*x^2-3*x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%K A077834 nonn,easy
%O A077834 0,2
%A A077834 _N. J. A. Sloane_, Nov 17 2002