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A192388 a(n) = A192387(n)/2.

%I A192388 #15 Jul 18 2023 04:14:19
%S A192388 0,1,2,16,48,256,928,4288,16896,73728,301056,1283072,5318656,22446080,
%T A192388 93634560,393543680,1646034944,6906380288,28918677504,121248940032,
%U A192388 507934408704,2129004593152,8920531730432,37385660268544,156658714017792
%N A192388 a(n) = A192387(n)/2.
%H A192388 G. C. Greubel, <a href="/A192388/b192388.txt">Table of n, a(n) for n = 1..1000</a>
%H A192388 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,12,-8,-16).
%F A192388 From _G. C. Greubel_, Jul 10 2023: (Start)
%F A192388 T(n, k) = [x^k] ((x+sqrt(x+5))^n - (x-sqrt(x+5))^n)/(2*sqrt(x+5)).
%F A192388 a(n) = (1/2)*Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1).
%F A192388 a(n) = 2*a(n-1) + 12*a(n-2) - 8*a(n-3) - 16*a(n-4).
%F A192388 G.f.: x^2/(1-2*x-12*x^2+8*x^3+16*x^4).
%F A192388 a(n) = 2^(n-1)*A112576(n). (End)
%e A192388 The first five polynomials p(n,x) and their reductions are as follows:
%e A192388   p(0,x) = 1 -> 1
%e A192388   p(1,x) =     2*x -> 2*x
%e A192388   p(2,x) = 3 +   x +  3*x^2 -> 8 + 4*x
%e A192388   p(3,x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 32*x
%e A192388   p(4,x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 96 + 96*x.
%e A192388 From these, read a(n) = (0, 2, 4, 32, 96, ...)/2 = (0, 1, 2, 16, 48, ...).
%t A192388 (* See A192386 *)
%t A192388 LinearRecurrence[{2,12,-8,-16}, {0,1,2,16}, 40] (* _G. C. Greubel_, Jul 10 2023 *)
%o A192388 (Magma)
%o A192388 R<x>:=PowerSeriesRing(Integers(), 41);
%o A192388 [0] cat Coefficients(R!( x^2/(1-2*x-12*x^2+8*x^3+16*x^4) )); // _G. C. Greubel_, Jul 10 2023
%o A192388 (SageMath)
%o A192388 @CachedFunction
%o A192388 def a(n): # a = A192388
%o A192388     if (n<5): return (0,0,1,2,16)[n]
%o A192388     else: return 2*a(n-1) +12*a(n-2) -8*a(n-3) -16*a(n-4)
%o A192388 [a(n) for n in range(1,41)] # _G. C. Greubel_, Jul 10 2023
%Y A192388 Cf. A112576, A192232, A192386, A192387.
%K A192388 nonn,easy
%O A192388 1,3
%A A192388 _Clark Kimberling_, Jun 30 2011