A192468 Constant term of the reduction by x^2->x+3 of the polynomial p(n,x)=1+x^n+x^(2n).

4, 16, 61, 304, 1546, 8107, 42748, 226240, 1198645, 6353944, 33688474, 178631251, 947215924, 5022815920, 26634734125, 141237718720, 748951245034, 3971518837243, 21060069709228, 111676816254688, 592197081386533, 3140288211876136

Offset

1,1

Comments

For an introduction to reductions of polynomials by substitutions such as x^2->x+3, see A192232.

Links

Table of n, a(n) for n=1..22.

Formula

Empirical G.f.: -x*(81*x^4-87*x^3-x^2+20*x-4)/((x-1)*(3*x^2+x-1)*(9*x^2-7*x+1)). [Colin Barker, Nov 12 2012]

Example

The first four polynomials p(n,x) and their reductions are as follows:
p(1,x)=1+x+x^2 -> 4+2x
p(2,x)=1+x^2+x^4 -> 16+8x
p(3,x)=1+x^3+x^6 -> 61+44x
p(4,x)=1+x^4+x^8 -> 304+224x.
From these, read
A192468=(4,16,61,304,...) and A192469=(2,8,44,224,...)

Mathematica

Remove["Global`*"];
q[x_] := x + 3; p[n_, x_] := 1 + x^n + x^(2 n);
Table[Simplify[p[n, x]], {n, 1, 5}]
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, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192468 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192469 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]
(* A192470 *)

Crossrefs

Cf. A192232, A192468, A192464, A192465.

Sequence in context: A206790 A283858 A283409 * A364921 A365835 A365860

Adjacent sequences: A192465 A192466 A192467 * A192469 A192470 A192471

Keyword

nonn

Author

Clark Kimberling, Jul 01 2011

Status

approved