A192344 Constant term of the reduction of n-th polynomial at A161516 by x^2->x+1.

1, 0, 5, 4, 49, 108, 637, 2024, 9329, 34104, 143621, 554092, 2255809, 8883876, 35708701, 141734480, 566950433, 2257038576, 9011796293, 35916665428, 143306508433, 571395546204, 2279250017533, 9089366457656, 36253101237521, 144581807030568

Offset

1,3

Comments

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)+10*a(n-2)-6*a(n-3)-9*a(n-4). G.f.: -(5*x^2+2*x-1) / (9*x^4+6*x^3-10*x^2-2*x+1). [Colin Barker, Nov 22 2012]

Example

The first four polynomials at A161516 and their reductions are as follows:
p0(x)=1 -> 1
p1(x)=x -> x
p2(x)=4+x+x^2 -> 5+2x
p3(x)=12x+3x^2+x^3 -> 4+17x.
From these, we read
A192344=(1,0,5,4,...) and A192345=(1,1,2,17...)

Mathematica

q[x_] := x + 1; d = Sqrt[x + 4];
p[n_, x_] := ((x + d)^n + (x - d)^n )/
  2 (* 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}]
(* A192344 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192345 *)

Crossrefs

Cf. A192232, A192345.

Sequence in context: A223527 A262421 A305170 * A181613 A264880 A128191

Adjacent sequences: A192341 A192342 A192343 * A192345 A192346 A192347

Keyword

nonn

Author

Clark Kimberling, Jun 28 2011

Status

approved