A246032 a(n) = A246031(2^n-1).

1, 26, 124, 1400, 10000, 89504, 707008, 5924480, 47900416, 393069824, 3189761536, 25963397888, 210468531712, 1706090904320, 13803141607936, 111595408530176, 901164713600512, 7271581998320384, 58625571435837952, 472335388734974720, 3803021424555945472, 30602681612309510912, 246127842107210007040

Offset: 0

Author

N. J. A. Sloane, Aug 16 2014; corrected Aug 21 2014

Keywords

• nonn

Comments

Comments from Michael Monagan on the computation of a(10) and a(11), Sep 01 2014: (Start)

I wrote a C program to compute them. Instead of storing monomials and coefficients, I just store monomials (presence of monomial means 1 mod 2) in an array - this saves a factor of 2 in space.

I used lexicographical order and packed the monomials in x,y,z into a 64 bit machine word: x^i y^j z^k is encoded as i*2^40+j*2^20+k. So the space needed to store p for n=10 is 3189761536 x 8 bytes = 25 gigs.

But the main gain is realizing that for the last step when we compute expand(p*g) mod 2, we don't need to save the product for the next iteration, so we just need to compute the number of terms in p*g mod 2 which we can do if we compute them in any monomial ordering without creating the product. (End)

Links

• Shalosh B. Ekhad, Details about A246031 and A246032
• Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
• Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
• N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
• N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A246031.

Programs

Magma

P := PolynomialRing(GF(2),3);g := (1+x+x^2)*(1+y+y^2)*(1+z+z^2)-x*y*z;
p := g;
for i := 2 to 9 do
  g := g*g;
  p := p*g;
  print(#Terms(p));
end for; // Roman Pearce, Aug 25 2014

Maple

# Maple program from N. J. A. Sloane, Aug 21 2014 with improvements from Roman Pearce, Aug 25 2014
# f is a 26-term polynomial, which describes a 3x3x3 cube with the center removed
f := expand((1+x+x^2)*(1+y+y^2)*(1+z+z^2)-x*y*z) mod 2;
# count nonzero terms in a polynomial
C := f->`if`(type(f,`+`),nops(f),1);
# Find number of ON cells in CA for generations 2^k-1 for k = 0..M
# defined by rule that cell is ON iff number of ON cells in nbd at
# time n-1 was odd where nbd is defined by a polynomial f(x, y, z).
OddCA2 := proc(f, M) global C; local n, a, i, g, p;
   g := expand(f) mod 2;
   p := g;
   a := [1,C(p)];
   map(lprint,a);
   for n from 2 to M do
     g := expand(g^2) mod 2;
     p := expand(p*g) mod 2;
     a := [op(a), C(p)];
     lprint(a[-1]);
   end do:
   [seq(a[i], i=1..nops(a))];
end proc:
OddCA2(f, 9);

Mathematica

f = PolynomialMod[(1+x+x^2)*(1+y+y^2)*(1+z+z^2) - x*y*z // Expand, 2];
c[f_] := If[f[[0]] === Plus, Length[f], 1];
OddCA2[f_, M_] := Module[{n, a, i, g, p},
  g = PolynomialMod[Expand[f], 2];
  p = g;
  a = {1, c[p]};
  Print[1]; Print[a[[-1]]];
  For[n = 2, n <= M, n++,
   g = PolynomialMod[Expand[g^2], 2];
   p = PolynomialMod[Expand[p*g], 2];
   a = Append[a, c[p]];
   Print[a[[-1]]]
  ];
a];
OddCA2[f, 9] (* Jean-François Alcover, Jan 20 2018, translated from Maple *)

Formula

The g.f. is

(1 + 6*x - 317*x^2 + 1718*x^3 + 5420*x^4 - 59432*x^5 + 61312*x^6 + 428928*x^7 - 887296*x^8 - 260096*x^9 + 737280*x^10)/((1 - 8*x)*(1 - 12*x - 17*x^2 + 608*x^3 - 856*x^4 - 9920*x^5 + 22576*x^6 + 52992*x^7 - 140032*x^8 - 29696*x^9 + 110592*x^10)),

found by Doron Zeilberger - see the Ekhad-Sloane-Zeilberger paper and the Ekhad link.

Extensions

a(7), a(8) and a(9) computed with Maple 18 and confirmed with MAGMA by Roman Pearce, Aug 25 2014

a(1)-a(9) confirmed by Michael Monagan, Aug 29 2014

a(10) and a(11) from Michael Monagan, Aug 29 2014

a(12) onwards from Doron Zeilberger, Feb 20 2015