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

%I A264292 #11 Dec 26 2021 05:29:59
%S A264292 0,0,1,2,4,7,15,26,55,101,221,413,870,1673,3490
%N A264292 Number of irreducible polynomials in the polynomial tree T generated as in Comments.
%C A264292 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.
%e A264292 First few generations:
%e A264292 g(0) = {0}
%e A264292 g(1) = {1}
%e A264292 g(2) = {2,x}
%e A264292 g(3) = {3, 2x, x+1, x^2}
%e A264292 g(4) = {4, 3x, 2x+1, 2x^2, x+2, x^2+x, x^2+1, x^3}
%e A264292 a(4) counts these 4 irreducible polynomials: 3x, 2x+1, x+2, x^2+1.
%t A264292 z = 15; t = Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {0}, z]];
%t A264292 s[0] = t[[1]]; s[n_] := s[n] = Union[t[[n]], s[n - 1]]
%t A264292 g[n_] := Complement[s[n], s[n - 1]]
%t A264292 Column[Table[g[z], {z, 1, 7}]]
%t A264292 Table[Count[Map[IrreduciblePolynomialQ, g[n]], True], {n, 1, z}]
%Y A264292 Cf. A000079, A262841.
%K A264292 nonn,easy
%O A264292 0,4
%A A264292 _Clark Kimberling_, Nov 24 2015