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

0,4

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.

Links

Table of n, a(n) for n=0..23.

Example

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)}

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 *)

Crossrefs

Cf. A000045, A264292.

Sequence in context: A265741 A254351 A346116 * A332070 A334741 A259973

Adjacent sequences: A262838 A262839 A262840 * A262842 A262843 A262844

Keyword

nonn,more

Author

Clark Kimberling, Nov 24 2015

Status

approved