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%I #14 Jan 03 2023 03:56:20
%S 0,-2,11,-52,247,-1182,5664,-27140,130037,-623044,2985181,-14302860,
%T 68529120,-328342742,1573184591,-7537580212,36114716467,-173036002122,
%U 829065294144,-3972290468600,19032387048857,-91189644775684,436915836829561,-2093389539372120
%N Expansion of -x*(2+x)/((1+x+x^2)*(1+5*x+x^2)).
%H Colin Barker, <a href="/A110308/b110308.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-6,-7,-6,-1).
%F a(n+2) = - 5*a(n+1) - a(n) - A099837(n+2).
%F a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - _Colin Barker_, Apr 30 2019
%F a(n) = (1/4)*(2*U(n, -5/2) + U(n-1, -5/2) - 2*U(n, -1/2) - U(n-1, -1/2)), where U(n, x) = ChebyshevU(n, x). - _G. C. Greubel_, Jan 03 2023
%p seriestolist(series(-x*(2+x)/((x^2+x+1)*(x^2+5*x+1)), x=0,25));
%t LinearRecurrence[{-6,-7,-6,-1}, {0,-2,11,-52}, 40] (* _G. C. Greubel_, Jan 03 2023 *)
%o (PARI) concat(0, Vec(-x*(2+x)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25))) \\ _Colin Barker_, Apr 30 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( -x*(2+x)/((1+x+x^2)*(1+5*x+x^2)) )); // _G. C. Greubel_, Jan 03 2023
%o (SageMath)
%o def U(n, x): return chebyshev_U(n,x)
%o def A110308(n): return (1/4)*(2*U(n, -5/2) +U(n-1, -5/2) -2*U(n, -1/2) -U(n-1, -1/2))
%o [A110308(n) for n in range(41)] # _G. C. Greubel_, Jan 03 2023
%Y Cf. A004253, A049347, A099837, A110307, A110309, A110310, A110311.
%K easy,sign
%O 0,2
%A _Creighton Dement_, Jul 19 2005
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