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.

1, 0, 10, 1, 137, 93, 2219, 3410, 39586, 94467, 750823, 2317249, 14833565, 53482716, 301162922, 1194377453, 6225350029, 26179063845, 130188268471, 567580989502, 2742763551458, 12225952022559, 58052436966875, 262325736910601

Offset: 1

Author

Clark Kimberling, Jun 29 2011

Keywords

• nonn

Comments

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

Crossrefs

Cf. A192232, A192358.

Programs

Mathematica

q[x_] := x + 1; d = 3;
p[n_, x_] := ((x + d)^n + (x - d)^n )/2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192357 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192358 *)

Formula

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]