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A192762 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192762 #14 Feb 19 2025 02:40:50
%S A192762 0,1,6,13,26,47,82,139,232,383,628,1025,1668,2709,4394,7121,11534,
%T A192762 18675,30230,48927,79180,128131,207336,335493,542856,878377,1421262,
%U A192762 2299669,3720962,6020663,9741658,15762355,25504048,41266439,66770524
%N A192762 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192762 The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+n+4 for n>0, where p(0,x)=1.  For discussions of polynomial reduction, see A192232 and A192744.
%H A192762 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A192762 a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). G.f.: x*(3*x^2-3*x-1) / ((x-1)^2*(x^2+x-1)). [_Colin Barker_, Dec 08 2012]
%t A192762 p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 4;
%t A192762 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192762 reduce[{p1_, q_, s_, x_}] :=
%t A192762 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192762        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192762 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192762 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192762   (* A022319 *)
%t A192762 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192762   (* A192762 *)
%Y A192762 Cf. A192744, A192232, A022319 (first differences).
%K A192762 nonn,easy
%O A192762 0,3
%A A192762 _Clark Kimberling_, Jul 09 2011