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A192748 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192748 #6 May 04 2014 16:52:09
%S A192748 0,1,4,11,24,47,86,151,258,433,718,1181,1932,3149,5120,8311,13476,
%T A192748 21835,35362,57251,92670,149981,242714,392761,635544,1028377,1663996,
%U A192748 2692451,4356528,7049063,11405678,18454831,29860602,48315529,78176230
%N A192748 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192748 The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+3n for n>0, where p(0,x)=1.  For discussions of polynomial reduction, see A192232 and A192744.
%F A192748 Conjecture: G.f.: -x^2*(1+x+x^2) / ( (x^2+x-1)*(x-1)^2 ), so the first differences are in A154691. - _R. J. Mathar_, May 04 2014
%t A192748 q = x^2; s = x + 1; z = 40;
%t A192748 p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 3 n;
%t A192748 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192748 reduce[{p1_, q_, s_, x_}] :=
%t A192748 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192748        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192748 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192748 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192748   (* A154691 *)
%t A192748 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192748   (* A192748 *)
%Y A192748 Cf. A192744, A192232.
%K A192748 nonn
%O A192748 1,3
%A A192748 _Clark Kimberling_, Jul 09 2011