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A099582 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1).

%I A099582 #11 Jul 22 2022 19:08:08
%S A099582 0,0,1,4,24,112,560,2688,13056,62976,304384,1469440,7096320,34263040,
%T A099582 165441536,798818304,3857055744,18623496192,89922273280,434183077888,
%U A099582 2096421666816,10122418978816,48875363631104,235991130439680
%N A099582 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1).
%C A099582 In general a(n) = Sum{k=0..floor(n/2)} binomial(n-k, k-1) * r^(n-k-1) has g.f. x^2/((1-r*x^2)*(1-r*x-r*x^2)) and satisfies a(n) = r*a(n-1) + 2*r*a(n-2) - r^2*a(n-3) - r^2*a(n-4).
%H A099582 G. C. Greubel, <a href="/A099582/b099582.txt">Table of n, a(n) for n = 0..1000</a>
%H A099582 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,8,-16,-16).
%F A099582 G.f.: x^2/((1-4*x^2)*(1-4*x-4*x^2)).
%F A099582 a(n) = 4*a(n-1) + 8*a(n-2) - 16*a(n-3) - 16*a(n-4) with a(n) = (n^3-n)/6 for n<5.
%F A099582 From _G. C. Greubel_, Jul 22 2022: (Start)
%F A099582 a(n) = 2^(n-4)*(2*A000129(n) - (1 - (-1)^n)).
%F A099582 a(n) = (1/4)*(A057087(n-1) - 2^(n-2)*(1 - (-1)^n)).
%F A099582 E.g.f.: (exp(2*x)*sinh(2*sqrt(2)*x) - sqrt(2)*sinh(2*x))/(8*sqrt(2)). (End)
%t A099582 Table[Sum[Binomial[n-k,k-1]*4^(n-k-1),{k,0,Floor[n/2]}],{n,0,30}] (* or *) LinearRecurrence[{4,8,-16,-16},{0,0,1,4},30] (* _Harvey P. Dale_, Jul 19 2012 *)
%o A099582 (Magma) I:=[0,0,1,4]; [n le 4 select I[n] else 4*(Self(n-1) +2*Self(n-2) -4*Self(n-3) -4*Self(n-4)): n in [1..41]]; // _G. C. Greubel_, Jul 22 2022
%o A099582 (SageMath) [2^(n-3)*(lucas_number1(n,2,-1) - (n%2)) for n in (0..40)] # _G. C. Greubel_, Jul 22 2022
%Y A099582 Cf. A099177, A099581.
%K A099582 easy,nonn
%O A099582 0,4
%A A099582 _Paul Barry_, Oct 23 2004