A192430 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

1, 1, 3, 9, 33, 113, 403, 1409, 4977, 17489, 61619, 216809, 763377, 2686881, 9458787, 33295297, 117206177, 412579681, 1452347043, 5112464521, 17996645761, 63350804881, 223004208243, 785007489729, 2763341973393, 9727369663793

Offset

0,3

Comments

The polynomial p(n,x) is defined by (u^n+v^n)//2)^n+(u^n-v^n)/(2d), where u=x+d, v=x-d, d=sqrt(x^2+2). For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

Links

Table of n, a(n) for n=0..25.

Formula

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -(x^3+5*x^2+x-1) / (x^4+2*x^3-6*x^2-2*x+1). - Colin Barker, May 12 2014

Example

The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=1+x -> 1+x
p(2,x)=2+3x+x^2 -> 3+4x
p(3,x)=2+7x+6x^2+x^3 -> 9+15x
p(4,x)=4+12x+17x^2+10x^3+x^4 -> 33+52x.
From these, read A192430=(1,1,3,9,33,...) and A192431=(0,1,4,15,52,...).

Mathematica

q[x_] := x + 1; d = Sqrt[x + 2];
u[x_] := x + d; v[x_] := x - d;
p[n_, x_] := (u[x]^n + v[x]^n)/2 + (u[x]^n - v[x]^n)/(2 d) (* A163762 *)
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[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192430 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192431 *)

Crossrefs

Cf. A192232, A192431.

Sequence in context: A037129 A148987 A176812 * A323790 A148988 A148989

Adjacent sequences: A192427 A192428 A192429 * A192431 A192432 A192433

Keyword

nonn

Author

Clark Kimberling, Jun 30 2011

Status

approved