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A099579 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1) * 3^(k-1).

%I A099579 #10 Jul 25 2022 01:13:37
%S A099579 0,0,1,1,7,10,40,70,217,427,1159,2440,6160,13480,32689,73129,173383,
%T A099579 392770,919480,2097790,4875913,11169283,25856071,59363920,137109280,
%U A099579 315201040,727060321,1672663441,3855438727,8873429050,20444528200
%N A099579 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1) * 3^(k-1).
%C A099579 In general a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*r^(k-1) has g.f. x^2/((1-r*x^2)*(1-x-r*x^2)) and satisfies the recurrence a(n) = a(n-1) + 2*r*a(n-2) - r*a(n-3) - r^2*a(n-4).
%H A099579 G. C. Greubel, <a href="/A099579/b099579.txt">Table of n, a(n) for n = 0..1000</a>
%H A099579 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-3,-9).
%F A099579 G.f.: x^2/((1-3*x^2)*(1-x-3*x^2)).
%F A099579 a(n) = a(n-1) + 6*a(n-2) - 3*a(n-3) - 9*a(n-4).
%F A099579 From _G. C. Greubel_, Jul 24 2022: (Start)
%F A099579 a(n) = (i*sqrt(3))^(n-1)*ChebyshevU(n-1, -i/(2*sqrt(3))) - 3^((n-1)/2)*(1 - (-1)^n)/2.
%F A099579 E.g.f.: (1/sqrt(39))*( 2*sqrt(3)*exp(x/2)*sinh(sqrt(13)*x/2) - sqrt(13)*sinh(sqrt(3)*x) ). (End)
%t A099579 LinearRecurrence[{1,6,-3,-9}, {0,0,1,1}, 50] (* _G. C. Greubel_, Jul 24 2022 *)
%o A099579 (Magma) [n le 4 select Floor((n-1)/2) else Self(n-1) +6*Self(n-2) -3*Self(n-3) -9*Self(n-4): n in [1..41]]; // _G. C. Greubel_, Jul 24 2022
%o A099579 (SageMath)
%o A099579 @CachedFunction
%o A099579 def a(n): # a = A099579
%o A099579     if (n<4): return (n//2)
%o A099579     else: return a(n-1) +6*a(n-2) -3*a(n-3) -9*a(n-4)
%o A099579 [a(n) for n in (0..40)] # _G. C. Greubel_, Jul 24 2022
%Y A099579 Cf. A097038, A099580, A140167.
%K A099579 easy,nonn
%O A099579 0,5
%A A099579 _Paul Barry_, Oct 23 2004