cp's OEIS Frontend

A192909 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

%I A192909 #14 Sep 08 2022 08:45:58
%S A192909 1,1,3,9,27,83,259,811,2541,7963,24957,78221,245165,768413,2408415,
%T A192909 7548629,23659463,74155215,232422687,728476151,2283243129,7156307287,
%U A192909 22429820697,70301181369,220343094521,690615411545,2164577236699
%N A192909 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.
%C A192909 The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x) + 1, with p(0,x) = 1, p(1,x) = x + 1.
%H A192909 G. C. Greubel, <a href="/A192909/b192909.txt">Table of n, a(n) for n = 0..1000</a>
%H A192909 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,1,0,-1).
%F A192909 a(n) = 4*a(n-1) - 3*a(n-2) + a(n-3) - a(n-5).
%F A192909 G.f.: (x^2-x+1)*(x^2+2*x-1) / ( (1-x)*(x^4+x^3+3*x-1) ). - _R. J. Mathar_, Jul 13 2011
%t A192909 u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 1; f = 1;
%t A192909 q = x^2; s = u*x + v; z = 24;
%t A192909 p[0, x_] := a;  p[1, x_] := b*x + c
%t A192909 p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
%t A192909 Table[Expand[p[n, x]], {n, 0, 8}]
%t A192909 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192909 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192909 u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192909 *)
%t A192909 u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192910 *)
%t A192909 Simplify[FindLinearRecurrence[u0]]
%t A192909 Simplify[FindLinearRecurrence[u1]]
%t A192909 LinearRecurrence[{4,-3,1,0,-1}, {1,1,3,9,27}, 30] (* _G. C. Greubel_, Jan 11 2019 *)
%o A192909 (PARI) my(x='x+O('x^30)); Vec((x^2-x+1)*(x^2+2*x-1)/((1-x)*(x^4+x^3+3*x -1))) \\ _G. C. Greubel_, Jan 11 2019
%o A192909 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (x^2-x+1)*(x^2+2*x-1)/((1-x)*(x^4+x^3+3*x-1)) )); // _G. C. Greubel_, Jan 11 2019
%o A192909 (Sage) ((x^2-x+1)*(x^2+2*x-1)/((1-x)*(x^4+x^3+3*x-1))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 11 2019
%o A192909 (GAP) a:=[1,1,3,9,27];; for n in [6..30] do a[n]:=4*a[n-1]-3*a[n-2] +a[n-3]-a[n-5]; od; a; # _G. C. Greubel_, Jan 11 2019
%Y A192909 Cf. A192232, A192744, A192872, A192904, A192910.
%K A192909 nonn
%O A192909 0,3
%A A192909 _Clark Kimberling_, Jul 12 2011