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 A253071 Number of odd terms in f^n, where f = 1/(x*y)+1/x+1/y+y+x/y+x+x*y. 2
 1, 7, 7, 21, 7, 49, 21, 95, 7, 49, 49, 147, 21, 147, 95, 333, 7, 49, 49, 147, 49, 343, 147, 665, 21, 147, 147, 441, 95, 665, 333, 1319, 7, 49, 49, 147, 49, 343, 147, 665, 49, 343, 343, 1029, 147, 1029, 665, 2331, 21, 147, 147, 441, 147, 1029, 441, 1995, 95, 665, 665, 1995, 333, 2331, 1319, 4837 (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 357 (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 A253072. EXAMPLE Here is the neighborhood f: [0, X, X] [X, 0, X] [X, X, X] which contains a(1) = 7 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*y)+1/x+1/y+y+x/y+x+x*y; OddCA(f, 130); MATHEMATICA (* f = A253072 *) f=1; f=7; f=21; f=95; f=333; f=1319; f[n_] := f[n] = -8 f[n-5] + 44 f[n-4] - 24 f[n-3] - 5 f[n-2] + 6 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, A252069. Cf. A253072. Sequence in context: A300091 A094248 A282828 * A283916 A146804 A147056 Adjacent sequences:  A253068 A253069 A253070 * A253072 A253073 A253074 KEYWORD nonn AUTHOR N. J. A. Sloane and Doron Zeilberger, Feb 23 2015 STATUS approved

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Last modified July 14 21:41 EDT 2020. Contains 335737 sequences. (Running on oeis4.)