A192346 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

1, 0, 3, 4, 25, 68, 275, 904, 3297, 11400, 40499, 141900, 500697, 1760396, 6200675, 21820432, 76823425, 270407696, 951914403, 3350807700, 11795463001, 41521535700, 146162319603, 514512119704, 1811159622625, 6375545788568, 22442862753875

Offset

1,3

Comments

To define the polynomials p(n,x), let d=sqrt(x+2); then p(n,x)=(1/2)((x+d)^n+(x-d)^n). These are similar to polynomials at A161516.

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

Links

Table of n, a(n) for n=1..27.

Formula

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -x*(x+1)*(3*x-1) / (x^4+2*x^3-6*x^2-2*x+1). [Colin Barker, Jan 17 2013]

Example

The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=2+x+x^2 -> 3+2x
p(3,x)=6x+3x^2+x^3 -> 4+11x.
From these, we read
A192346=(1,0,3,4,...) and A192347=(1,1,2,11...)

Mathematica

q[x_] := x + 1; d = Sqrt[x + 2];
p[n_, x_] := ((x + d)^n + (x - d)^n )/ 2
(* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 4}]
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}]
(* A192346 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192347 *)

Crossrefs

Cf. A192232, A192347.

Sequence in context: A245244 A009391 A212696 * A335739 A055348 A004206

Adjacent sequences: A192343 A192344 A192345 * A192347 A192348 A192349

Keyword

nonn

Author

Clark Kimberling, Jun 28 2011

Status

approved