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%I #16 Jan 04 2023 02:02:14
%S 1,-4,17,-80,384,-1842,8827,-42292,202631,-970862,4651680,-22287540,
%T 106786021,-511642564,2451426797,-11745491420,56276030304,
%U -269634660102,1291897270207,-6189851690932,29657361184451,-142096954231322,680827409972160,-3262040095629480
%N Expansion of (1+2*x)/((1+x+x^2)*(1+5*x+x^2)).
%H Colin Barker, <a href="/A110307/b110307.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+1).
%F a(n) + a(n+1) + a(n+2) = A002320(n).
%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)*(3*U(n,-5/2) + U(n-1,-5/2) + 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((1+2*x)/((x^2+x+1)*(x^2+5*x+1)), x=0,25));
%t LinearRecurrence[{-6,-7,-6,-1}, {1,-4,17,-80}, 41] (* _G. C. Greubel_, Jan 03 2023 *)
%o (PARI) Vec((1+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(), 30); Coefficients(R!( (1+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 A110307(n): return (1/4)*(3*U(n,-5/2) +U(n-1,-5/2) +U(n,-1/2) -U(n-1,-1/2))
%o [A110307(n) for n in range(41)] # _G. C. Greubel_, Jan 03 2023
%Y Cf. A002320, A004253, A049347, A099837, A110308, A110309, A110310, A110311.
%K easy,sign
%O 0,2
%A _Creighton Dement_, Jul 19 2005
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