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A192801 Constant term in the reduction of the polynomial (x+2)^n by x^3->x^2+x+1. See Comments.

%I A192801 #18 Feb 18 2025 12:49:21
%S A192801 1,2,4,9,25,84,312,1199,4637,17906,68976,265249,1019069,3913484,
%T A192801 15026092,57690143,221487945,850350482,3264725772,12534190569,
%U A192801 48122302705,184755243892,709328262928,2723314511871,10455585321989,40141990468066
%N A192801 Constant term in the reduction of the polynomial (x+2)^n by x^3->x^2+x+1.  See Comments.
%C A192801 For discussions of polynomial reduction, see A192232 and A192744.
%C A192801 If the same reduction is applied to the sequence (x+1)^n instead of (x+2)^n, the resulting three coefficient sequences are essentially as follows:
%C A192801 A078484: constants
%C A192801 A099216: coefficients of x
%C A192801 A115390: coefficients of x^2.
%H A192801 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-15,11).
%F A192801 a(n) = 7*a(n-1)-15*a(n-2)+11*a(n-3).
%F A192801 G.f.: -(5*x^2-5*x+1)/(11*x^3-15*x^2+7*x-1). [_Colin Barker_, Jul 27 2012]
%e A192801 The first five polynomials p(n,x) and their reductions:
%e A192801 p(1,x)=1 -> 1
%e A192801 p(2,x)=x+2 -> x+2
%e A192801 p(3,x)=x^2+4x+4 -> x^2+1
%e A192801 p(4,x)=x^3+6x^2+12x+8 -> x^2+4x+4
%e A192801 p(5,x)=x^4+8x^3+24x^2+32x+16 -> 7x^2+13*x+9, so that
%e A192801 A192798=(1,2,4,9,25,...), A192799=(0,1,4,13,42,...), A192800=(0,0,1,7,34,...).
%t A192801 q = x^3; s = x^2 + x + 1; z = 40;
%t A192801 p[n_, x_] := (x + 2)^n;
%t A192801 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192801 reduce[{p1_, q_, s_, x_}] :=
%t A192801 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192801        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192801 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192801 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]  (* A192801 *)
%t A192801 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192801   (* A192802 *)
%t A192801 u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
%t A192801   (* A192803 *)
%Y A192801 Cf. A192744, A192232, A192616, A192802, A192803.
%K A192801 nonn,easy
%O A192801 0,2
%A A192801 _Clark Kimberling_, Jul 10 2011
%E A192801 Recurrence corrected by _Colin Barker_, Jul 27 2012