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A192752 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192752 #8 May 04 2014 17:34:16
%S A192752 1,7,12,23,39,66,109,179,292,475,771,1250,2025,3279,5308,8591,13903,
%T A192752 22498,36405,58907,95316,154227,249547,403778,653329,1057111,1710444,
%U A192752 2767559,4478007,7245570,11723581,18969155,30692740,49661899,80354643
%N A192752 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192752 The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+4n+3 for n>0, where p(0,x)=1.  For discussions of polynomial reduction, see A192232 and A192744.
%F A192752 Conjecture: G.f.: ( 1+5*x-2*x^2 ) / ( (x-1)*(x^2+x-1) ). a(n) = A000071(n+3)+5*A000071(n+2) -2*A000071(n+1) and first differences in A022136. - _R. J. Mathar_, May 04 2014
%t A192752 q = x^2; s = x + 1; z = 40;
%t A192752 p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 4 n + 3;
%t A192752 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192752 reduce[{p1_, q_, s_, x_}] :=
%t A192752 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192752        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192752 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192752 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}](* A192752 *)
%t A192752 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}](* A192753 *)
%Y A192752 Cf. A192744, A192232, A192753.
%K A192752 nonn
%O A192752 0,2
%A A192752 _Clark Kimberling_, Jul 09 2011