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 A253069 Number of odd terms in f^n, where f = 1/x+1+x+x/y+y/x+x*y. 2
 1, 6, 6, 22, 6, 36, 22, 82, 6, 36, 36, 132, 22, 132, 82, 302, 6, 36, 36, 132, 36, 216, 132, 492, 22, 132, 132, 484, 82, 492, 302, 1106, 6, 36, 36, 132, 36, 216, 132, 492, 36, 216, 216, 792, 132, 792, 492, 1812, 22, 132, 132, 484, 132, 792, 484, 1804, 82, 492, 492, 1804, 302, 1812, 1106, 4066 (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 175 (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, 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, 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, 2015 FORMULA This is the Run Length Transform of A253070. EXAMPLE Here is the neighborhood f: [X, 0, X] [X, X, X] [0, 0, X] 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+x/y+y/x+x*y; OddCA(f, 130); MATHEMATICA (* f = A253070 *) f=1; f=6; f=22; f=82; f=302; f=1106; f=4066; 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&], {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, A253066. Cf. A253070. Sequence in context: A188273 A185786 A178822 * A255460 A255464 A325998 Adjacent sequences:  A253066 A253067 A253068 * A253070 A253071 A253072 KEYWORD nonn AUTHOR N. J. A. Sloane, Jan 29 2015 STATUS approved

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Last modified October 21 14:56 EDT 2019. Contains 328301 sequences. (Running on oeis4.)