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A192906 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

%I A192906 #15 Sep 08 2022 08:45:58
%S A192906 1,1,2,7,23,72,225,705,2210,6927,21711,68048,213281,668481,2095202,
%T A192906 6566935,20582567,64511384,202196289,633738369,1986309058,6225634847,
%U A192906 19512839199,61158565024
%N A192906 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.
%C A192906 The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x + 1.
%H A192906 G. C. Greubel, <a href="/A192906/b192906.txt">Table of n, a(n) for n = 0..1000</a>
%H A192906 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,1,1).
%F A192906 a(n) = 3*a(n-1) + a(n-3) + a(n-4).
%F A192906 G.f.: (1-2*x-x^2)/(1-3*x-x^3-x^4). - _Colin Barker_, Aug 31 2012
%t A192906 (* To obtain very general results, delete the next line. *)
%t A192906 u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 1; f = 0;
%t A192906 q = x^2; s = u*x + v; z = 24;
%t A192906 p[0, x_] := a;
%t A192906 p[1, x_] := b*x + c; p[n_, x_] :=  d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f; Table[Expand[p[n, x]], {n, 0, 8}] (* p(0,x), p(1,x), ... p(5,x) *)
%t A192906 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192906 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192906 u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]    (* A192904 *)
%t A192906 u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]    (* A192905 *)
%t A192906 Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *)
%t A192906 Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *)
%t A192906 LinearRecurrence[{3,0,1,1}, {1,1,2,7}, 30] (* _G. C. Greubel_, Jan 11 2019 *)
%o A192906 (PARI) my(x='x+O('x^30)); Vec((1-2*x-x^2)/(1-3*x-x^3-x^4)) \\ _G. C. Greubel_, Jan 11 2019
%o A192906 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-2*x-x^2)/(1-3*x-x^3-x^4) )); // _G. C. Greubel_, Jan 11 2019
%o A192906 (Sage) ((1-2*x-x^2)/(1-3*x-x^3-x^4)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 11 2019
%o A192906 (GAP) a:=[1,1,2,7];; for n in [5..30] do a[n]:=3*a[n-1]+a[n-3]+a[n-4]; od; a; # _G. C. Greubel_, Jan 11 2019
%Y A192906 Cf. A192232, A192744, A192872, A192904, A192907.
%K A192906 nonn,easy
%O A192906 0,3
%A A192906 _Clark Kimberling_, Jul 12 2011