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A246039
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Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y)+1.
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2
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1, 7, 7, 29, 7, 49, 29, 103, 7, 49, 49, 203, 29, 203, 103, 373, 7, 49, 49, 203, 49, 343, 203, 721, 29, 203, 203, 841, 103, 721, 373, 1407, 7, 49, 49, 203, 49, 343, 203, 721, 49, 343, 343, 1421, 203, 1421, 721, 2611, 29, 203, 203, 841, 203, 1421, 841, 2987, 103, 721, 721, 2987, 373, 2611, 1407, 5277
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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 575 (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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EXAMPLE
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Here is the neighborhood:
[X, X, X]
[0, X, 0]
[X, X, X]
which contains a(1) = 7 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)+1 mod 2;
OddCA(f, 70);
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MATHEMATICA
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(* f = A246038 *) f[0]=1; f[1]=7; f[2]=29; f[3]=103; f[4]=373; f[n_] := f[n] = 8 f[n-4] + 8 f[n-3] + 3 f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[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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