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A192357 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+3)^n+(x-3)^n) by x^2->x+1.

%I A192357 #7 Nov 22 2012 08:04:54
%S A192357 1,0,10,1,137,93,2219,3410,39586,94467,750823,2317249,14833565,
%T A192357 53482716,301162922,1194377453,6225350029,26179063845,130188268471,
%U A192357 567580989502,2742763551458,12225952022559,58052436966875,262325736910601
%N A192357 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+3)^n+(x-3)^n) by x^2->x+1.
%C A192357 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F A192357 Conjecture: a(n) = 2*a(n-1)+19*a(n-2)-20*a(n-3)-55*a(n-4). G.f.: x*(x^3-9*x^2-2*x+1)/((5*x^2+5*x+1)*(11*x^2-7*x+1)). [_Colin Barker_, Nov 22 2012]
%t A192357 q[x_] := x + 1; d = 3;
%t A192357 p[n_, x_] := ((x + d)^n + (x - d)^n )/2 (* similar to polynomials defined at A161516 *)
%t A192357 Table[Expand[p[n, x]], {n, 0, 6}]
%t A192357 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192357    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192357 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
%t A192357 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
%t A192357 (* A192357 *)
%t A192357 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
%t A192357 (* A192358 *)
%Y A192357 Cf. A192232, A192358.
%K A192357 nonn
%O A192357 1,3
%A A192357 _Clark Kimberling_, Jun 29 2011