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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 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: A177967 A265741 A254351 * A259973 A092362 A105766

Adjacent sequences:  A262838 A262839 A262840 * A262842 A262843 A262844

KEYWORD

nonn,easy,more

AUTHOR

Clark Kimberling, Nov 24 2015

STATUS

approved

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Last modified April 29 22:12 EDT 2017. Contains 285615 sequences.