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A264293 Number of irreducible polynomials in the n-th generation of polynomials generated as in Comments. 0
0, 0, 2, 4, 9, 20, 54, 131, 354, 912, 2457, 6429, 17081, 44850, 118578, 311471 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..15.

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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}]

CROSSREFS

Cf. A001906, A262841, A264292.

Sequence in context: A002861 A032200 A130969 * A034750 A079060 A225173

Adjacent sequences:  A264290 A264291 A264292 * A264294 A264295 A264296

KEYWORD

nonn,more

AUTHOR

Clark Kimberling, Nov 24 2015

EXTENSIONS

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

STATUS

approved

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Last modified March 22 17:25 EDT 2019. Contains 321422 sequences. (Running on oeis4.)