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 A253064 Number of odd terms in f^n, where f = 1/x+1+x+y. 6
 1, 4, 4, 12, 4, 16, 12, 40, 4, 16, 16, 48, 12, 48, 40, 128, 4, 16, 16, 48, 16, 64, 48, 160, 12, 48, 48, 144, 40, 160, 128, 416, 4, 16, 16, 48, 16, 64, 48, 160, 16, 64, 64, 192, 48, 192, 160, 512, 12, 48, 48, 144, 48, 192, 144, 480, 40, 160, 160, 480, 128, 512, 416 (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 017 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 25 2015 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 A087206. EXAMPLE Here is the neighborhood f: [0, X, 0] [X, X, X] which contains a(1) = 4 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+y; OddCA(f, 130); MATHEMATICA f[n_] := 2^n*Fibonacci[n+2]; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[] == 1&], {n, 0, 62}] (* Jean-François Alcover, Jul 11 2017 *) CROSSREFS Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035. Cf. A087206. Sequence in context: A256261 A256251 A256139 * A109045 A079315 A229253 Adjacent sequences:  A253061 A253062 A253063 * A253065 A253066 A253067 KEYWORD nonn AUTHOR N. J. A. Sloane, Jan 26 2015 STATUS approved

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Last modified October 15 12:31 EDT 2019. Contains 328026 sequences. (Running on oeis4.)