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%I #13 Jan 03 2023 02:07:10
%S 1,-7,36,-173,827,-3960,18973,-90907,435564,-2086913,9998999,
%T -47908080,229541401,-1099798927,5269453236,-25247467253,120967883027,
%U -579591947880,2776991856373,-13305367333987,63749844813564,-305443856733833,1463469438855599,-7011903337544160
%N Expansion of (1-x+x^2)/((x^2+x+1)*(x^2+5*x+1)).
%H Colin Barker, <a href="/A110310/b110310.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) - (-1)^n*A109265(n+3).
%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) = (1/2)*(3*ChevyshevU(n, -5/2) - ChebyshevU(n, -1/2)). - _G. C. Greubel_, Jan 02 2023
%p seriestolist(series((1-x+x^2)/((x^2+x+1)*(x^2+5*x+1)), x=0,25));
%t LinearRecurrence[{-6,-7,-6,-1}, {1,-7,36,-173}, 40] (* _G. C. Greubel_, Jan 02 2023 *)
%o (PARI) Vec((1-x+x^2)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ _Colin Barker_, Apr 30 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^2)/((1+x+x^2)*(1+5*x+x^2)) )); // _G. C. Greubel_, Jan 02 2023
%o (SageMath)
%o def U(n,x): return chebyshev_U(n,x)
%o def A110310(n): return (1/2)*(3*U(n, -5/2) - U(n, -1/2))
%o [A110310(n) for n in range(41)] # _G. C. Greubel_, Jan 02 2023
%Y Cf. A004253, A049347, A109265, A110307, A110308, A110309, A110311.
%K easy,sign
%O 0,2
%A _Creighton Dement_, Jul 19 2005
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