A192350 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

1, 0, 6, 4, 64, 128, 896, 2752, 14208, 52224, 238592, 946176, 4110336, 16830464, 71598080, 297140224, 1253048320, 5229707264, 21973303296, 91924463616, 385642135552, 1614916091904, 6770569248768, 28364203098112, 118885634277376

Offset

1,3

Comments

To define the polynomials p(n,x), let d=sqrt(x+5); 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..25.

Formula

Conjecture: a(n) = 2*a(n-1)+12*a(n-2)-8*a(n-3)-16*a(n-4). G.f.: -x*(6*x^2+2*x-1) / (16*x^4+8*x^3-12*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)=5+x+x^2 -> 6+2x
p(3,x)=15x+3x^2+x^3 -> 4+20x.
From these, we read
A192350=(1,0,6,4,...) and A192351=(0,1,2,20...)

Mathematica

q[x_] := x + 1; d = Sqrt[x + 5];
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}]
  (* A192350 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
  (* A192351 *)

Crossrefs

Cf. A192232, A192351.

Sequence in context: A327370 A260716 A112521 * A308900 A354949 A239861

Adjacent sequences: A192347 A192348 A192349 * A192351 A192352 A192353

Keyword

nonn

Author

Clark Kimberling, Jun 28 2011

Status

approved