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A253066
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Number of odd terms in f^n, where f = 1/x+1+x+1/y+y/x+x*y.
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5
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1, 6, 6, 28, 6, 36, 28, 112, 6, 36, 36, 168, 28, 168, 112, 456, 6, 36, 36, 168, 36, 216, 168, 672, 28, 168, 168, 784, 112, 672, 456, 1816, 6, 36, 36, 168, 36, 216, 168, 672, 36, 216, 216, 1008, 168, 1008, 672, 2736, 28, 168, 168, 784, 168, 1008, 784, 3136, 112, 672, 672, 3136, 456, 2736, 1816, 7288
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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 275 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
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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 A253068.
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EXAMPLE
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Here is the neighborhood f:
[X, 0, X]
[X, X, X]
[0, X, 0]
which contains a(1) = 6 ON cells.
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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+y/x+x*y;
OddCA(f, 130);
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MATHEMATICA
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(* f = A253068 *) f[0] = 1; f[n_] := ((-2)^n + 4^(n+2)-8)/9; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1 &], {n, 0, 63}] (* Jean-François Alcover, Jul 12 2017 *)
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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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