cp's OEIS Frontend

A192348 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

%I A192348 #7 Jan 17 2013 09:05:16
%S A192348 1,0,4,4,36,88,432,1408,5776,20736,80320,297792,1132096,4242304,
%T A192348 16028928,60276736,227287296,855703552,3224482816,12144337920,
%U A192348 45752574976,172339107840,649223532544,2445572276224,9212566081536,34703459811328
%N A192348 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.
%C A192348 To define the polynomials p(n,x), let d=sqrt(x+3); then p(n,x)=(1/2)((x+d)^n+(x-d)^n).  These are similar to polynomials at A161516.
%C A192348 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F A192348 Conjecture: a(n) = 2*a(n-1)+8*a(n-2)-4*a(n-3)-4*a(n-4). G.f.: -x*(4*x^2+2*x-1) / (4*x^4+4*x^3-8*x^2-2*x+1). [_Colin Barker_, Jan 17 2013]
%e A192348 The first four polynomials p(n,x) and their reductions are as follows:
%e A192348 p(0,x)=1 -> 1
%e A192348 p(1,x)=x -> x
%e A192348 p(2,x)=3+x+x^2 -> 4+2x
%e A192348 p(3,x)=9x+3x^2+x^3 -> 4+14x.
%e A192348 From these, we read
%e A192348 A192348=(1,0,3,4,...) and A192349=(0,1,2,14...)
%t A192348 q[x_] := x + 1; d = Sqrt[x + 3];
%t A192348 p[n_, x_] := ((x + d)^n + (x - d)^n )/
%t A192348   2 (* similar to polynomials defined at A161516 *)
%t A192348 Table[Expand[p[n, x]], {n, 0, 4}]
%t A192348 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192348    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192348 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
%t A192348 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
%t A192348 (* A192348 *)
%t A192348 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
%t A192348 (* A192349 *)
%Y A192348 Cf. A192232, A192349.
%K A192348 nonn
%O A192348 1,3
%A A192348 _Clark Kimberling_, Jun 28 2011