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A192424 a(n) = A192423(n)/2.

%I A192424 #9 Jul 12 2023 05:53:03
%S A192424 1,0,2,1,8,10,39,70,208,439,1162,2640,6641,15600,38362,91481,222688,
%T A192424 534650,1295559,3119990,7544888,18194639,43958642,106072320,256167361,
%U A192424 618303360,1492941842,3603915601,8701212248,21005629450
%N A192424 a(n) = A192423(n)/2.
%H A192424 G. C. Greubel, <a href="/A192424/b192424.txt">Table of n, a(n) for n = 0..1000</a>
%H A192424 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-1,-1).
%F A192424 From _G. C. Greubel_, Jul 11 2023: (Start)
%F A192424 a(n) = (1/2)*Sum_{j=0..n} T(n, j)*A078008(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n.
%F A192424 a(n) = (1/3)*((-1)^n*A000032(n) + A000129(n+1) - A000129(n)).
%F A192424 a(n) = a(n-1) + 4*a(n-2) - a(n-3) - a(n-4).
%F A192424 G.f.: (1+x)*(1-2*x)/((1+x-x^2)*(1-2*x-x^2)). (End)
%e A192424 The first five polynomials p(n,x) and their reductions are as follows:
%e A192424   p(0,x) = 2 -> 2
%e A192424   p(1,x) = x -> x
%e A192424   p(2,x) = 2 + x^2 -> 4 + x
%e A192424   p(3,x) = 3*x + x^3 -> 2 + 6*x
%e A192424   p(4,x) = 2 + 4*x^2 + x^4 -> 16 + 9*x.
%e A192424 From these, read A192423(n) = 2*a(n) = (2, 0, 4, 2, 16, ...) and A192425 = (0, 1, 1, 6, 9, ...).
%t A192424 (See A192423.)
%t A192424 LinearRecurrence[{1,4,-1,-1}, {1,0,2,1}, 40] (* _G. C. Greubel_, Jul 12 2023 *)
%o A192424 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-2*x)/((1+x-x^2)*(1-2*x-x^2)) )); // _G. C. Greubel_, Jul 12 2023
%o A192424 (SageMath)
%o A192424 @CachedFunction
%o A192424 def a(n): # a = A192424
%o A192424     if (n<4): return (1,0,2,1)[n]
%o A192424     else: return a(n-1) +4*a(n-2) -a(n-3) -a(n-4)
%o A192424 [a(n) for n in range(41)] # _G. C. Greubel_, Jul 12 2023
%Y A192424 Cf. A000032, A000129, A078008, A161514, A192232, A192423, A192425.
%K A192424 nonn,easy
%O A192424 0,3
%A A192424 _Clark Kimberling_, Jun 30 2011