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A291337 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 2 S - 2 S^3.

%I A291337 #9 Jun 02 2023 21:52:10
%S A291337 1,3,10,34,115,387,1300,4366,14665,49263,165490,555934,1867555,
%T A291337 6273687,21075220,70798066,237832225,798950763,2683918570,9016098634,
%U A291337 30287816995,101745987387,341795711140,1148195728966,3857138603785,12957301471863,43527515777650
%N A291337 p-INVERT of (1,1,1,1,1,...), where p(S) = 1 - 2 S - 2 S^3.
%H A291337 Clark Kimberling, <a href="/A291337/b291337.txt">Table of n, a(n) for n = 0..1000</a>
%H A291337 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,5).
%F A291337 G.f.: (1 - 2*x + 2*x^2)/(1 - 5*x + 7*x^2 - 5*x^3).
%F A291337 a(n) = 5*a(n-1) - 7*a(n-2) + 5*a(n-3) for n >= 4.
%F A291337 a(n) = (1/2)*A291005(n).
%t A291337 z = 60; s = 1 - 2 s - 2 s^3;
%t A291337 Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A000012 *)
%t A291337 u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291005 *)
%t A291337 u / 2  (* A291337 *)
%o A291337 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x+2*x^2)/(1-5*x+7*x^2-5*x^3) )); // _G. C. Greubel_, Jun 01 2023
%o A291337 (SageMath)
%o A291337 def A291337_list(prec):
%o A291337     P.<x> = PowerSeriesRing(ZZ, prec)
%o A291337     return P( (1-2*x+2*x^2)/(1-5*x+7*x^2-5*x^3) ).list()
%o A291337 A291337_list(30) # _G. C. Greubel_, Jun 01 2023
%Y A291337 Cf. A000012, A289780, A291000, A291005.
%K A291337 nonn,easy
%O A291337 0,2
%A A291337 _Clark Kimberling_, Aug 23 2017