A264293 Number of irreducible polynomials in the n-th generation of polynomials generated as in Comments.

0, 0, 2, 4, 9, 20, 54, 131, 354, 912, 2457, 6429, 17081, 44850, 118578, 311471

Offset: 0

Author

Clark Kimberling, Nov 24 2015

Keywords

• nonn
• more

Comments

The set of polynomials 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 and y*p is in T. The n-th generation of T consists of F(2n) polynomials, for n >= 0, where F = A000045 = Fibonacci numbers.

Note that a given polynomial can appear only once; e.g., though x*y can arise either from multiplying x by y or y by x, it occurs only once in generation 3. Also although 0*x = 0, 0 occurs only in generation 0. - Robert Israel, Nov 22 2018

Examples

			First few generations:
g(0) = {0}
g(1) = {1}
g(2) = {2,x,y}
g(3) = {3, 2x, x^2, 1+x, 2y, xy, y^2, 1+y}
The irreducible polynomials in g(3) are 2x, 1+x, 2y, 1+y, so that a(3) = 4.
		

Crossrefs

Cf. A001906, A262841, A264292.

Programs

Maple

A[0]:= 0: A[1]:= 0:
T:= {1}:
for n from 2 to 13 do
  T:= map(t -> (t+1,expand(x*t),expand(y*t)),T);
  A[n]:= nops(select(irreduc,T));
od:
seq(A[i],i=0..13); # Robert Israel, Nov 22 2018

Mathematica

z = 12; t = Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#, y*#} &, #], 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]]
Table[Length[g[z]], {z, 1, z}]
Column[Table[g[z], {z, 1, 6}]]
Table[Count[Map[IrreduciblePolynomialQ, g[n]], True], {n, 1, z}]

Extensions

Edited, and a(12)-a(15) from Robert Israel, Nov 22 2018