A262841 Number of irreducible polynomials occurring as the first component of a vertex in the Fibonacci zero tree, generated as in Comments.

0, 0, 1, 2, 3, 5, 8, 11, 21, 28, 54, 68, 135, 183, 360, 470, 948, 1234, 2479, 3294, 6531, 8713, 17120, 23200

Offset: 0

Author

Clark Kimberling, Nov 24 2015

Keywords

• nonn
• more

Comments

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 n-th generation of T is F(n+1), where F = A000045, the Fibonacci numbers, for n >= 0.

The number of irreducible polynomials occurring as the second component of a vertex in the tree T is a(n-1) for n >= 1.

Examples

			First few generations:
g(0) = {(0,0)}
g(1) = {(0,2), (1,0)}
g(2) = {(0,3), (2,0), (0,x)}
g(3) = {(0,4), (3,0), (0,2x), (0,1+x), (x,0)}
g(4) = {(0,5), (4,0), (0,3x), (0,1+2x), (2x,0), (0,2+x), (1+x,0), (0,x^2)}
		

Crossrefs

Cf. A000045, A264292.

Programs

Mathematica

z = 20; g = {{{0, 0}}};
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 = Table[Drop[g[[k + 1]], Length[g[[k]]]], {k, Length[g] - 1}];
Map[Length, t] (* Fibonacci numbers *)
Map[Count[IrreduciblePolynomialQ[#], {_, True}] &, t]
(* Peter J. C. Moses, Oct 19 2015 *)