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 A253066 Number of odd terms in f^n, where f = 1/x+1+x+1/y+y/x+x*y. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..8191 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 [math.CO], 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 [math.CO], 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 [math.CO], 2015. FORMULA This is the Run Length Transform of A253068. EXAMPLE Here is the neighborhood f: [X, 0, X] [X, X, X] [0, X, 0] which contains a(1) = 6 ON cells. MAPLE 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); MATHEMATICA (* f = A253068 *) f = 1; f[n_] := ((-2)^n + 4^(n+2)-8)/9; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[] == 1 &], {n, 0, 63}] (* Jean-François Alcover, Jul 12 2017 *) CROSSREFS Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035, A253064, A253065. Cf. A253068. Sequence in context: A241865 A243122 A274940 * A016725 A267651 A151779 Adjacent sequences:  A253063 A253064 A253065 * A253067 A253068 A253069 KEYWORD nonn AUTHOR N. J. A. Sloane, Jan 29 2015 STATUS approved

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Last modified March 23 18:13 EDT 2019. Contains 321433 sequences. (Running on oeis4.)