A192465 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x)=1+x^n+x^(2n).

3, 9, 25, 93, 353, 1389, 5505, 21933, 87553, 349869, 1398785, 5593773, 22372353, 89483949, 357924865, 1431677613, 5726666753, 22906579629, 91626143745, 366504225453, 1466016202753, 5864063412909, 23456250855425, 93824997829293

Offset

1,1

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..24.

Formula

Empirical G.f.: -x*(3*x-1)*(8*x^2-3)/((x-1)*(x+1)*(2*x-1)*(4*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 -> 3+2x
p(2,x)=1+x^2+x^4 -> 9+6x
p(3,x)=1+x^3+x^6 -> 25+24x
p(4,x)=1+x^4+x^8 -> 93+90x.
From these, read
A192465=(3,9,25,93,...) and A192466=(2,6,24,90,...)

Mathematica

Remove["Global`*"];
q[x_] := x + 2; 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}]
(* A192465 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192466 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]
(* A192467 *)

Crossrefs

Cf. A192232, A192466, A192464.

Sequence in context: A246653 A192371 A363467 * A012771 A351891 A178061

Adjacent sequences: A192462 A192463 A192464 * A192466 A192467 A192468

Keyword

nonn

Author

Clark Kimberling, Jul 01 2011

Status

approved