%I
%S 0,0,1,2,3,5,8,11,21,28,54,68,135,183,360,470,948,1234,2479,3294,6531,
%T 8713,17120,23200
%N Number of irreducible polynomials occurring as the first component of a vertex in the Fibonacci zero tree, generated as in Comments.
%C The tree T, which we call the Fibonacci zero tree, is generated by these rules: (0, 0) is in T, and if (0, h) is in T, then (0, h + 1) is in T, and if (k, 0) is in T, then (0, k*x) is in T. The number of vertices (f(x),g(x)) in the nth generation of T is F(n+1), where F = A000045, the Fibonacci numbers, for n >= 0.
%C The number of irreducible polynomials occurring as the second component of a vertex in the tree T is a(n1) for n >= 1.
%e First few generations:
%e g(0) = {(0,0)}
%e g(1) = {(0,2), (1,0)}
%e g(2) = {(0,3), (2,0), (0,x)}
%e g(3) = {(0,4), (3,0), (0,2x), (0,1+x), (x,0)}
%e g(4) = {(0,5), (4,0), (0,3x), (0,1+2x), (2x,0), (0,2+x), (1+x,0), (0,x^2)}
%t z = 20; g = {{{0, 0}}};
%t Do[AppendTo[g, DeleteDuplicates[Partition[Flatten[Join[g, Map[# /. {{0, k_} > {{0, k + 1}, {k, 0}}, {k_, 0} > {0, x*k}} &, g]]], 2]]], {z}]
%t t = Table[Drop[g[[k + 1]], Length[g[[k]]]], {k, Length[g]  1}];
%t Map[Length, t] (* Fibonacci numbers *)
%t Map[Count[IrreduciblePolynomialQ[#], {_, True}] &, t]
%t (* _Peter J. C. Moses_, Oct 19 2015 *)
%Y Cf. A000045, A264292.
%K nonn,more
%O 0,4
%A _Clark Kimberling_, Nov 24 2015
