A255477 -id:A255477 - cp's OEIS frontend

A255478 The subsequence A255477(2^n-1).

1, 80, 624, 16480, 221888, 4245888, 66759424, 1158095360, 18967114752, 318729757696, 5260906022912, 87309457787904, 1439402906245120, 23729936541559808, 389928297670440960, 6398667538798480384, 104773970517568231424, 1712967144350639066112, 27958316901583753934848

Offset: 0

Doron Zeilberger, Feb 26 2015

nonn

Shalosh B. Ekhad, Details about A255477 and A255478
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, 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, 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, 2015
Index entries for sequences related to cellular automata

Cf. A255477.

For g.f. see Ekhad link.

A246031 Number of ON cells in 3-D cellular automaton described in Comments, after n generations.

Original entry on oeis.org

1, 26, 26, 124, 26, 676, 124, 1400, 26, 676, 676, 3224, 124, 3224, 1400, 10000, 26, 676, 676, 3224, 676, 17576, 3224, 36400, 124, 3224, 3224, 15376, 1400, 36400, 10000, 89504, 26, 676, 676, 3224, 676, 17576, 3224, 36400, 676, 17576, 17576, 83824, 3224, 83824, 36400, 260000, 124, 3224, 3224, 15376, 3224, 83824, 15376, 173600, 1400, 36400, 36400, 173600, 10000, 260000, 89504, 707008

Offset: 0

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

nonn

We work on the cells of the 3-D grid. Each cell has 26 neighbors, A cell is ON iff an odd number of its neighbors were ON at the previous generation. We start with a single ON cell.

The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g., 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).

			The entries form blocks of sizes 1,1,2,4,8,...:
1,
26,
26, 124,
26, 676, 124, 1400,
26, 676, 676, 3224, 124, 3224, 1400, 10000,
26, 676, 676, 3224, 676, 17576, 3224, 36400, 124, 3224, 3224, 15376, 1400, 36400, 10000, 89504,
26, 676, 676, 3224, 676, 17576, 3224, 36400, 676, 17576, 17576, 83824, 3224, 83824, 36400, 260000, 124, 3224, 3224, 15376, 3224, 83824, 15376, 173600, 1400, 36400, 36400, 173600, 10000, 260000, 89504, 707008
...
From _Omar E. Pol_, Mar 19 2015: (Start)
Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below:
1;
..
26;
...
26;
124;
..........
26,   676;
124;
1400;
.....................
26,   676, 676, 3224;
124,  3224;
1400;
10000;
............................................
26,   676,  676, 3224, 676,17576,3224,36400;
124,  3224, 3224, 15376;
1400, 36400;
10000;
89504;
..........................................................................................
26,   676,  676, 3224, 676,17576,3224,36400,676,17576,17576,83824,3224,83824,36400,260000;
124,  3224, 3224, 15376, 3224, 83824, 15376, 173600;
1400, 36400, 36400, 173600;
10000, 260000;
89504;
707008;
...
Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1022
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, 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, 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, 2015
Index entries for sequences related to cellular automata

A 3-D analog of A160239 (2-D) and A255477 (4-D). Cf. A246032.

P := PolynomialRing(GF(2),3);
f := (1+x+x^2)*(1+y+y^2)*(1+z+z^2)-x*y*z;
p := 1;
for i := 1 to 100 do
  p := p*f;
  print(#Terms(p));
end for; // Roman Pearce, Feb 18 2015

# This is a very inefficient program!
f:=expand((1+x+x^2)*(1+y+y^2)*(1+z+z^2))-x*y*z;
g:=n->expand(f^n) mod 2;
h:=n->subs({x=1,y=1,z=1},g(n));
[seq(h(n),n=0..30)];
# Better program from Roman Pearce, Feb 18 2015:
f := Expand((1+x+x^2)*(1+y+y^2)*(1+z+z^2)-x*y*z) mod 2:
p := 1;
for i from 1 to 100 do
  p := Expand(p*f) mod 2;
  lprint(nops(p));
end do:

f = (1 + x + x^2)*(1 + y + y^2)*(1 + z + z^2) - x*y*z;
p = 1; Print[1];
Join[{1}, Table[p = Expand[p*f] // PolynomialMod[#, 2]&; Lp = Length[p]; Print[Lp]; Lp, 100]] (* Jean-François Alcover, Jan 17 2018 *)

This is the Run Length Transform of A246032 (see Comments).

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A255478 The subsequence A255477(2^n-1).

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A246031 Number of ON cells in 3-D cellular automaton described in Comments, after n generations.

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