A264292 - OEIS

%I #11 Dec 26 2021 05:29:59

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

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

%C 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 First few generations:

%e g(0) = {0}

%e g(1) = {1}

%e g(2) = {2,x}

%e g(3) = {3, 2x, x+1, x^2}

%e g(4) = {4, 3x, 2x+1, 2x^2, x+2, x^2+x, x^2+1, x^3}

%e a(4) counts these 4 irreducible polynomials: 3x, 2x+1, x+2, x^2+1.

%t z = 15; t = Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {0}, z]];

%t s[0] = t[[1]]; s[n_] := s[n] = Union[t[[n]], s[n - 1]]

%t g[n_] := Complement[s[n], s[n - 1]]

%t Column[Table[g[z], {z, 1, 7}]]

%t Table[Count[Map[IrreduciblePolynomialQ, g[n]], True], {n, 1, z}]

%Y Cf. A000079, A262841.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Nov 24 2015