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A246031 Number of ON cells in 3-D cellular automaton described in Comments, after n generations. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

LINKS

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

FORMULA

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

EXAMPLE

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)

MAPLE

# 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:

PROG

// MAGMA program from Roman Pearce, Feb 18 2015:

P<x, y, z> := 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;

CROSSREFS

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

Sequence in context: A003900 A040651 A022360 * A165847 A217981 A094835

Adjacent sequences:  A246028 A246029 A246030 * A246032 A246033 A246034

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified June 26 23:21 EDT 2017. Contains 288777 sequences.