A192471 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^n+x^(2n+1).

2, 5, 10, 24, 59, 150, 386, 1001, 2606, 6800, 17767, 46458, 121538, 318045, 832418, 2178920, 5703875, 14931950, 39090754, 102338337, 267921062, 701419680, 1836329615, 4807555634, 12586315394, 32951355125, 86267692666, 225851630136

Offset: 1

Author

Clark Kimberling, Jul 01 2011

Keywords

• nonn

Comments

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Examples

			The first four polynomials p(n,x) and their reductions are as follows:
p(1,x)=1+x+x^3 -> 2+3x
p(2,x)=1+x^2+x^5 -> 5+6x
p(3,x)=1+x^3+x^7 -> 10+15x
p(4,x)=1+x^4+x^9 -> 24+37x.
From these, read
A192471=(2,5,10,24,...) and A087124=(3,6,15,37,...)
		

Crossrefs

Cf. A192232.

Programs

Mathematica

Remove["Global`*"];
q[x_] := x + 1; p[n_, x_] := 1 + x^n + x^(2 n+1);
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}]
(* A192471 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A087124 *)

Formula

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