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A192385 a(n) = A192384(n)/2.

%I A192385 #9 Jul 10 2023 08:21:19
%S A192385 0,1,2,12,36,156,544,2144,7872,30096,112416,425536,1598528,6031296,
%T A192385 22699008,85552128,322177024,1213849856,4572111360,17224104960,
%U A192385 64880993280,244410981376,920685043712,3468237545472,13064787542016
%N A192385 a(n) = A192384(n)/2.
%H A192385 G. C. Greubel, <a href="/A192385/b192385.txt">Table of n, a(n) for n = 1..1000</a>
%H A192385 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,8,-4,-4).
%F A192385 From _G. C. Greubel_, Jul 10 2023: (Start)
%F A192385 T(n, k) = [x^k] ((x+sqrt(x+3))^n - (x-sqrt(x+3))^n)/(2*sqrt(x+3)).
%F A192385 a(n) = (1/2)*Sum_{k=0..n-1} T(n, k)*Fibonacci(k).
%F A192385 a(n) = 2*a(n-1) + 8*a(n-2) - 4*a(n-3) - 4*a(n-4).
%F A192385 G.f.: x^2/(1 - 2*x - 8*x^2 + 4*x^3 + 4*x^4). (End)
%t A192385 (See A192384.)
%t A192385 LinearRecurrence[{2,8,-4,-4}, {0,1,2,12}, 40] (* _G. C. Greubel_, Jul 10 2023 *)
%o A192385 (Magma)
%o A192385 R<x>:=PowerSeriesRing(Integers(), 41);
%o A192385 [0] cat Coefficients(R!( x^2/(1-2*x-8*x^2+4*x^3+4*x^4) )); // _G. C. Greubel_, Jul 10 2023
%o A192385 (SageMath)
%o A192385 @CachedFunction
%o A192385 def a(n): # a = A192385
%o A192385     if (n<5): return (0,0,1,2,12)[n]
%o A192385     else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4)
%o A192385 [a(n) for n in range(1,41)] # _G. C. Greubel_, Jul 10 2023
%Y A192385 Cf. A192232, A192383, A192384.
%K A192385 nonn
%O A192385 1,3
%A A192385 _Clark Kimberling_, Jun 30 2011