cp's OEIS Frontend

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

%I A264293 #16 Nov 24 2018 01:28:19
%S A264293 0,0,2,4,9,20,54,131,354,912,2457,6429,17081,44850,118578,311471
%N A264293 Number of irreducible polynomials in the n-th generation of polynomials generated as in Comments.
%C A264293 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.
%C A264293 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
%e A264293 First few generations:
%e A264293 g(0) = {0}
%e A264293 g(1) = {1}
%e A264293 g(2) = {2,x,y}
%e A264293 g(3) = {3, 2x, x^2, 1+x, 2y, xy, y^2, 1+y}
%e A264293 The irreducible polynomials in g(3) are 2x, 1+x, 2y, 1+y, so that a(3) = 4.
%p A264293 A[0]:= 0: A[1]:= 0:
%p A264293 T:= {1}:
%p A264293 for n from 2 to 13 do
%p A264293   T:= map(t -> (t+1,expand(x*t),expand(y*t)),T);
%p A264293   A[n]:= nops(select(irreduc,T));
%p A264293 od:
%p A264293 seq(A[i],i=0..13); # _Robert Israel_, Nov 22 2018
%t A264293 z = 12; t = Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#, y*#} &, #], 1]] &, {0}, z]];
%t A264293 s[0] = t[[1]]; s[n_] := s[n] = Union[t[[n]], s[n - 1]]
%t A264293 g[n_] := Complement[s[n], s[n - 1]]
%t A264293 Table[Length[g[z]], {z, 1, z}]
%t A264293 Column[Table[g[z], {z, 1, 6}]]
%t A264293 Table[Count[Map[IrreduciblePolynomialQ, g[n]], True], {n, 1, z}]
%Y A264293 Cf. A001906, A262841, A264292.
%K A264293 nonn,more
%O A264293 0,3
%A A264293 _Clark Kimberling_, Nov 24 2015
%E A264293 Edited, and a(12)-a(15) from _Robert Israel_, Nov 22 2018