A192348 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

1, 0, 4, 4, 36, 88, 432, 1408, 5776, 20736, 80320, 297792, 1132096, 4242304, 16028928, 60276736, 227287296, 855703552, 3224482816, 12144337920, 45752574976, 172339107840, 649223532544, 2445572276224, 9212566081536, 34703459811328

Offset

1,3

Comments

To define the polynomials p(n,x), let d=sqrt(x+3); 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..26.

Formula

Conjecture: a(n) = 2*a(n-1)+8*a(n-2)-4*a(n-3)-4*a(n-4). G.f.: -x*(4*x^2+2*x-1) / (4*x^4+4*x^3-8*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)=3+x+x^2 -> 4+2x
p(3,x)=9x+3x^2+x^3 -> 4+14x.
From these, we read
A192348=(1,0,3,4,...) and A192349=(0,1,2,14...)

Mathematica

q[x_] := x + 1; d = Sqrt[x + 3];
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}]
(* A192348 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192349 *)

Crossrefs

Cf. A192232, A192349.

Sequence in context: A335183 A129357 A100303 * A111882 A321313 A070959

Adjacent sequences: A192345 A192346 A192347 * A192349 A192350 A192351

Keyword

nonn

Author

Clark Kimberling, Jun 28 2011

Status

approved