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A099784 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * 2^k * (-2)^(n-3*k).

%I A099784 #10 Sep 08 2022 08:45:15
%S A099784 1,-2,4,-6,4,16,-92,312,-848,1960,-3824,5760,-3824,-15392,88384,
%T A099784 -299616,814144,-1881344,3669568,-5524608,3657472,14807680,-84909824,
%U A099784 287723520,-781639424,1805843968,-3521371136,5298829824
%N A099784 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * 2^k * (-2)^(n-3*k).
%C A099784 In general a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * u^k * v^(n-3*k) has g.f. (1-v*x)/((1-v*x)^2 - u*x^2) and satisfies the recurrence a(n) = 2*u*v*a(n-1) - v^2*a(n-2) + u*a(n-3).
%H A099784 G. C. Greubel, <a href="/A099784/b099784.txt">Table of n, a(n) for n = 0..1000</a>
%H A099784 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-4,-4,2).
%F A099784 G.f.: (1+2*x)/((1+2*x)^2 - 2*x^3).
%F A099784 a(n) = Sum_{k=0..floor(n/3)} C(n-k, 2*k)*2^(n-2*k)*(-1)^(n-3*k).
%F A099784 a(n) = -4*a(n-1) - 4*a(n-2) + 2*a(n-3).
%p A099784 seq(coeff(series((1-2*x)/((1-2*x)^2 - 2*x^3), x, n+1), x, n), n = 0 .. 30); # _G. C. Greubel_, Sep 04 2019
%t A099784 LinearRecurrence[{-4,-4,2}, {1,-2,4}, 30] (* _G. C. Greubel_, Sep 04 2019 *)
%o A099784 (PARI) my(x='x+O('x^30)); Vec((1-2*x)/((1-2*x)^2 - 2*x^3)) \\ _G. C. Greubel_, Sep 04 2019
%o A099784 (Magma) I:=[1,-2,4]; [n le 3 select I[n] else -4*Self(n-1) - 4*Self(n-2) + 2*Self(n-3): n in [1..30]]; // _G. C. Greubel_, Sep 04 2019
%o A099784 (Sage)
%o A099784 def A099784_list(prec):
%o A099784     P.<x> = PowerSeriesRing(ZZ, prec)
%o A099784     return P((1-2*x)/((1-2*x)^2 - 2*x^3)).list()
%o A099784 A099784_list(30) # _G. C. Greubel_, Sep 04 2019
%o A099784 (GAP) a:=[1,-2,4];; for n in [4..30] do a[n]:=-4*a[n-1]-4*a[n-2] + 2*a[n-3]; od; a; # _G. C. Greubel_, Sep 04 2019
%Y A099784 Cf. A099780, A099781, A099782, A099783, A099785, A099786, A099787.
%K A099784 easy,sign
%O A099784 0,2
%A A099784 _Paul Barry_, Oct 26 2004