A264292 -id:A264292 - cp's OEIS frontend

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

Clark Kimberling, Nov 24 2015

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.

			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)}

Cf. A000045, A264292.

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 *)

A264293 Number of irreducible polynomials in the n-th generation of polynomials generated as in Comments.

Original entry on oeis.org

0, 0, 2, 4, 9, 20, 54, 131, 354, 912, 2457, 6429, 17081, 44850, 118578, 311471

Offset: 0

Clark Kimberling, Nov 24 2015

The set of polynomials T is generated by these rules: 0 is in T, and if p is in T, then p + 1 is in T and x*p is in T and y*p is in T. The n-th generation of T consists of F(2n) polynomials, for n >= 0, where F = A000045 = Fibonacci numbers.

Note that a given polynomial can appear only once; e.g., though x*y can arise either from multiplying x by y or y by x, it occurs only once in generation 3. Also although 0*x = 0, 0 occurs only in generation 0. - Robert Israel, Nov 22 2018

			First few generations:
g(0) = {0}
g(1) = {1}
g(2) = {2,x,y}
g(3) = {3, 2x, x^2, 1+x, 2y, xy, y^2, 1+y}
The irreducible polynomials in g(3) are 2x, 1+x, 2y, 1+y, so that a(3) = 4.

Cf. A001906, A262841, A264292.

A[0]:= 0: A[1]:= 0:
T:= {1}:
for n from 2 to 13 do
  T:= map(t -> (t+1,expand(x*t),expand(y*t)),T);
  A[n]:= nops(select(irreduc,T));
od:
seq(A[i],i=0..13); # Robert Israel, Nov 22 2018

z = 12; t = Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#, y*#} &, #], 1]] &, {0}, z]];
s[0] = t[[1]]; s[n_] := s[n] = Union[t[[n]], s[n - 1]]
g[n_] := Complement[s[n], s[n - 1]]
Table[Length[g[z]], {z, 1, z}]
Column[Table[g[z], {z, 1, 6}]]
Table[Count[Map[IrreduciblePolynomialQ, g[n]], True], {n, 1, z}]

Edited, and a(12)-a(15) from Robert Israel, Nov 22 2018

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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.

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