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Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.
3

%I #14 Sep 08 2022 08:45:58

%S 1,1,3,9,27,83,259,811,2541,7963,24957,78221,245165,768413,2408415,

%T 7548629,23659463,74155215,232422687,728476151,2283243129,7156307287,

%U 22429820697,70301181369,220343094521,690615411545,2164577236699

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

%C 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 G. C. Greubel, <a href="/A192909/b192909.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,1,0,-1).

%F a(n) = 4*a(n-1) - 3*a(n-2) + a(n-3) - a(n-5).

%F 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 u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 1; f = 1;

%t q = x^2; s = u*x + v; z = 24;

%t p[0, x_] := a; p[1, x_] := b*x + c

%t p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;

%t Table[Expand[p[n, x]], {n, 0, 8}]

%t reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

%t u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192909 *)

%t u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192910 *)

%t Simplify[FindLinearRecurrence[u0]]

%t Simplify[FindLinearRecurrence[u1]]

%t LinearRecurrence[{4,-3,1,0,-1}, {1,1,3,9,27}, 30] (* _G. C. Greubel_, Jan 11 2019 *)

%o (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 (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 (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 (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 Cf. A192232, A192744, A192872, A192904, A192910.

%K nonn

%O 0,3

%A _Clark Kimberling_, Jul 12 2011