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A099580 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1) * 4^(k-1).

%I A099580 #13 Jul 24 2022 04:25:09
%S A099580 0,0,1,1,9,13,65,117,441,909,2929,6565,19305,45565,126881,309141,
%T A099580 833049,2069613,5467345,13745797,35877321,90860509,235418369,
%U A099580 598860405,1544728185,3940169805,10135859761,25896538981,66507086889,170093242813
%N A099580 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1) * 4^(k-1).
%C A099580 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 A099580 G. C. Greubel, <a href="/A099580/b099580.txt">Table of n, a(n) for n = 0..1000</a>
%H A099580 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,8,-4,-16).
%F A099580 G.f.: x^2/((1-4*x^2) * (1-x-4*x^2)).
%F A099580 a(n) = a(n-1) + 8*a(n-2) - 4*a(n-3) - 16*a(n-4).
%F A099580 From _G. C. Greubel_, Jul 24 2022: (Start)
%F A099580 a(n) = (4*(2/i)^(n-1)*ChebyshevU(n-1, i/4) - 2^n*(1-(-1)^n))/4.
%F A099580 E.g.f.: ( 4*exp(x/2)*sinh(sqrt(17)*x/2) - sqrt(17)*sinh(2*x) )/(2*sqrt(17)). (End)
%t A099580 LinearRecurrence[{1,8,-4,-16}, {0,0,1,1}, 51] (* _G. C. Greubel_, Jul 24 2022 *)
%o A099580 (Magma) [n le 4 select Floor((n-1)/2) else Self(n-1) +8*Self(n-2) -4*Self(n-3) -16*Self(n-4): n in [1..41]]; // _G. C. Greubel_, Jul 24 2022
%o A099580 (SageMath)
%o A099580 @CachedFunction
%o A099580 def a(n): # a = A099580
%o A099580     if (n<4): return (n//2)
%o A099580     else: return a(n-1) +8*a(n-2) -4*a(n-3) -16*a(n-4)
%o A099580 [a(n) for n in (0..40)] # _G. C. Greubel_, Jul 24 2022
%Y A099580 Cf. A006131, A097038, A099579.
%K A099580 easy,nonn
%O A099580 0,5
%A A099580 _Paul Barry_, Oct 23 2004