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%I #13 Jan 03 2023 02:07:31
%S 1,-3,12,-57,275,-1320,6325,-30303,145188,-695637,3332999,-15969360,
%T 76513801,-366599643,1756484412,-8415822417,40322627675,-193197315960,
%U 925663952125,-4435122444663,21249948271188,-101814618911277,487823146285199,-2337301112514720
%N Expansion of (1+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)).
%H Colin Barker, <a href="/A110309/b110309.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)*(ChebyshevU(n, -5/2) + ChebyshevU(n, -1/2)). - _G. C. Greubel_, Jan 03 2023
%p seriestolist(series((1+3*x+x^2)/((x^2+5*x+1)*(x^2+x+1)), x=0,25));
%t LinearRecurrence[{-6,-7,-6,-1}, {1,-3,12,-57}, 40] (* _G. C. Greubel_, Jan 03 2023 *)
%o (PARI) Vec((1+3*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+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)) )); // _G. C. Greubel_, Jan 03 2023
%o (SageMath)
%o def A110309(n): return (1/2)*(chebyshev_U(n,-5/2)+chebyshev_U(n,-1/2))
%o [A110309(n) for n in range(41)] # _G. C. Greubel_, Jan 03 2023
%Y Cf. A004253, A049347, A109265, A110307, A110308, A110310, A110311.
%K easy,sign
%O 0,2
%A _Creighton Dement_, Jul 19 2005
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