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A192346 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

%I A192346 #9 Jan 17 2013 09:05:23
%S A192346 1,0,3,4,25,68,275,904,3297,11400,40499,141900,500697,1760396,6200675,
%T A192346 21820432,76823425,270407696,951914403,3350807700,11795463001,
%U A192346 41521535700,146162319603,514512119704,1811159622625,6375545788568,22442862753875
%N A192346 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.
%C A192346 To define the polynomials p(n,x), let d=sqrt(x+2); then p(n,x)=(1/2)((x+d)^n+(x-d)^n).  These are similar to polynomials at A161516.
%C A192346 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F A192346 Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -x*(x+1)*(3*x-1) / (x^4+2*x^3-6*x^2-2*x+1). [_Colin Barker_, Jan 17 2013]
%e A192346 The first four polynomials p(n,x) and their reductions are as follows:
%e A192346 p(0,x)=1 -> 1
%e A192346 p(1,x)=x -> x
%e A192346 p(2,x)=2+x+x^2 -> 3+2x
%e A192346 p(3,x)=6x+3x^2+x^3 -> 4+11x.
%e A192346 From these, we read
%e A192346 A192346=(1,0,3,4,...) and A192347=(1,1,2,11...)
%t A192346 q[x_] := x + 1; d = Sqrt[x + 2];
%t A192346 p[n_, x_] := ((x + d)^n + (x - d)^n )/ 2
%t A192346 (* similar to polynomials defined at A161516 *)
%t A192346 Table[Expand[p[n, x]], {n, 0, 4}]
%t A192346 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192346    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192346 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
%t A192346 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
%t A192346 (* A192346 *)
%t A192346 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
%t A192346 (* A192347 *)
%Y A192346 Cf. A192232, A192347.
%K A192346 nonn
%O A192346 1,3
%A A192346 _Clark Kimberling_, Jun 28 2011