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A192468 Constant term of the reduction by x^2->x+3 of the polynomial p(n,x)=1+x^n+x^(2n). 3
4, 16, 61, 304, 1546, 8107, 42748, 226240, 1198645, 6353944, 33688474, 178631251, 947215924, 5022815920, 26634734125, 141237718720, 748951245034, 3971518837243, 21060069709228, 111676816254688, 592197081386533, 3140288211876136 (list; graph; refs; listen; history; text; internal format)
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 * A276042 A084859 A081666

Adjacent sequences:  A192465 A192466 A192467 * A192469 A192470 A192471

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 01 2011

STATUS

approved

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Last modified September 27 08:39 EDT 2021. Contains 347689 sequences. (Running on oeis4.)