

A262841


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


3



0, 0, 1, 2, 3, 5, 8, 11, 21, 28, 54, 68, 135, 183, 360, 470, 948, 1234, 2479, 3294, 6531, 8713, 17120, 23200
(list;
graph;
refs;
listen;
history;
text;
internal format)



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 nth 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(n1) 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: A177967 A265741 A254351 * A259973 A092362 A105766
Adjacent sequences: A262838 A262839 A262840 * A262842 A262843 A262844


KEYWORD

nonn,more,changed


AUTHOR

Clark Kimberling, Nov 24 2015


STATUS

approved



