%I #7 Nov 22 2012 08:04:54

%S 1,0,10,1,137,93,2219,3410,39586,94467,750823,2317249,14833565,

%T 53482716,301162922,1194377453,6225350029,26179063845,130188268471,

%U 567580989502,2742763551458,12225952022559,58052436966875,262325736910601

%N 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 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

%F 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 q[x_] := x + 1; d = 3;

%t p[n_, x_] := ((x + d)^n + (x - d)^n )/2 (* similar to polynomials defined at A161516 *)

%t Table[Expand[p[n, x]], {n, 0, 6}]

%t reductionRules = {x^y_?EvenQ -> q[x]^(y/2),

%t x^y_?OddQ -> x q[x]^((y - 1)/2)};

%t t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]

%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]

%t (* A192357 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]

%t (* A192358 *)

%Y Cf. A192232, A192358.

%K nonn

%O 1,3

%A _Clark Kimberling_, Jun 29 2011