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A192430 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192430 #7 May 12 2014 13:06:02
%S A192430 1,1,3,9,33,113,403,1409,4977,17489,61619,216809,763377,2686881,
%T A192430 9458787,33295297,117206177,412579681,1452347043,5112464521,
%U A192430 17996645761,63350804881,223004208243,785007489729,2763341973393,9727369663793
%N A192430 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192430 The polynomial p(n,x) is defined by (u^n+v^n)//2)^n+(u^n-v^n)/(2d), where u=x+d, v=x-d, d=sqrt(x^2+2).  For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.
%F A192430 Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -(x^3+5*x^2+x-1) / (x^4+2*x^3-6*x^2-2*x+1). - _Colin Barker_, May 12 2014
%e A192430 The first five polynomials p(n,x) and their reductions are as follows:
%e A192430 p(0,x)=1 -> 1
%e A192430 p(1,x)=1+x -> 1+x
%e A192430 p(2,x)=2+3x+x^2 -> 3+4x
%e A192430 p(3,x)=2+7x+6x^2+x^3 -> 9+15x
%e A192430 p(4,x)=4+12x+17x^2+10x^3+x^4 -> 33+52x.
%e A192430 From these, read A192430=(1,1,3,9,33,...) and A192431=(0,1,4,15,52,...).
%t A192430 q[x_] := x + 1; d = Sqrt[x + 2];
%t A192430 u[x_] := x + d; v[x_] := x - d;
%t A192430 p[n_, x_] := (u[x]^n + v[x]^n)/2 + (u[x]^n - v[x]^n)/(2 d) (* A163762 *)
%t A192430 Table[Expand[p[n, x]], {n, 0, 6}]
%t A192430 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192430 t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 0, 30}]
%t A192430 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192430 *)
%t A192430 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192431 *)
%Y A192430 Cf. A192232, A192431.
%K A192430 nonn
%O A192430 0,3
%A A192430 _Clark Kimberling_, Jun 30 2011