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A246035
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Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y).
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13
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1, 9, 9, 25, 9, 81, 25, 121, 9, 81, 81, 225, 25, 225, 121, 441, 9, 81, 81, 225, 81, 729, 225, 1089, 25, 225, 225, 625, 121, 1089, 441, 1849, 9, 81, 81, 225, 81, 729, 225, 1089, 81, 729, 729, 2025, 225, 2025, 1089, 3969, 25, 225, 225, 625, 225, 2025, 625, 3025, 121, 1089, 1089, 3025, 441, 3969, 1849, 7225, 9, 81, 81, 225, 81, 729, 225
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OFFSET
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0,2
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COMMENTS
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This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.
This is the odd-rule cellular automaton defined by OddRule 777 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
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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EXAMPLE
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Here is the neighborhood:
[X, X, X]
[X, X, X]
[X, X, X]
which contains a(1) = 9 ON cells.
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From Omar E. Pol, Mar 17 2015: (Start)
Apart from the initial 1, the sequence can be written also as an irregular tetrahedron T(s,r,k) = A139818(r+2) * a(k), s>=1, 1<=r<=s, 0<=k<=(A011782(s-r)-1) as shown below:
..
9;
...
9;
25;
..........
9, 81;
25;
121;
....................
9, 81, 81, 225;
25, 225;
121;
441;
........................................
9, 81, 81, 225, 81, 729, 225, 1089;
25, 225, 225, 625;
121, 1089;
441;
1849;
...
Note that every row r is equal to A139818(r+2) times the beginning of the sequence itself, thus in 3D every column contains the same number: T(s,r,k) = T(s+1,r,k).
(End)
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MAPLE
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C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru 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 or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=(1/x+1+x)*(1/y+1+y);
OddCA(f, 70);
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MATHEMATICA
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b[0] = 1; b[n_] := b[n] = Expand[b[n - 1]*(x^2 + x + 1)];
a[n_] := Count[CoefficientList[b[n], x], _?OddQ]^2;
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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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