A264292 Number of irreducible polynomials in the polynomial tree T generated as in Comments.

0, 0, 1, 2, 4, 7, 15, 26, 55, 101, 221, 413, 870, 1673, 3490

Offset: 0

Author

Clark Kimberling, Nov 24 2015

Keywords

• nonn
• easy

Comments

The tree T is generated by these rules: 0 is in T, and if p is in T, then p + 1 is in T and x*p is in T. Every polynomial with nonnegative integer coefficients is in T, and the n-th generation of T consists of 2^(n-1) polynomials, for n >= 1.

Examples

			First few generations:
g(0) = {0}
g(1) = {1}
g(2) = {2,x}
g(3) = {3, 2x, x+1, x^2}
g(4) = {4, 3x, 2x+1, 2x^2, x+2, x^2+x, x^2+1, x^3}
a(4) counts these 4 irreducible polynomials: 3x, 2x+1, x+2, x^2+1.
		

Crossrefs

Cf. A000079, A262841.

Programs

Mathematica

z = 15; t = Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {0}, z]];
s[0] = t[[1]]; s[n_] := s[n] = Union[t[[n]], s[n - 1]]
g[n_] := Complement[s[n], s[n - 1]]
Column[Table[g[z], {z, 1, 7}]]
Table[Count[Map[IrreduciblePolynomialQ, g[n]], True], {n, 1, z}]