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A246031
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Number of ON cells in 3-D cellular automaton described in Comments, after n generations.
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4
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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
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OFFSET
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0,2
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COMMENTS
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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).
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LINKS
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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
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FORMULA
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This is the Run Length Transform of A246032 (see Comments).
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EXAMPLE
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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
...
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)
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MAPLE
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# 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)];
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:
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MATHEMATICA
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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 *)
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PROG
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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;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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