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A192350 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

%I A192350 #7 Jan 17 2013 09:05:09
%S A192350 1,0,6,4,64,128,896,2752,14208,52224,238592,946176,4110336,16830464,
%T A192350 71598080,297140224,1253048320,5229707264,21973303296,91924463616,
%U A192350 385642135552,1614916091904,6770569248768,28364203098112,118885634277376
%N A192350 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.
%C A192350 To define the polynomials p(n,x), let d=sqrt(x+5); then p(n,x)=(1/2)((x+d)^n+(x-d)^n).  These are similar to polynomials at A161516.
%C A192350 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F A192350 Conjecture: a(n) = 2*a(n-1)+12*a(n-2)-8*a(n-3)-16*a(n-4). G.f.: -x*(6*x^2+2*x-1) / (16*x^4+8*x^3-12*x^2-2*x+1). [_Colin Barker_, Jan 17 2013]
%e A192350 The first four polynomials p(n,x) and their reductions are as follows:
%e A192350 p(0,x)=1 -> 1
%e A192350 p(1,x)=x -> x
%e A192350 p(2,x)=5+x+x^2 -> 6+2x
%e A192350 p(3,x)=15x+3x^2+x^3 -> 4+20x.
%e A192350 From these, we read
%e A192350 A192350=(1,0,6,4,...) and A192351=(0,1,2,20...)
%t A192350 q[x_] := x + 1; d = Sqrt[x + 5];
%t A192350 p[n_, x_] := ((x + d)^n + (x - d)^n )/
%t A192350   2 (* similar to polynomials defined at A161516 *)
%t A192350 Table[Expand[p[n, x]], {n, 0, 4}]
%t A192350 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192350    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192350 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
%t A192350 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
%t A192350   (* A192350 *)
%t A192350 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
%t A192350   (* A192351 *)
%Y A192350 Cf. A192232, A192351.
%K A192350 nonn
%O A192350 1,3
%A A192350 _Clark Kimberling_, Jun 28 2011