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 A246039 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y)+1. 2
 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 (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 575 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). Run Length Transform of A246038. 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). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..8192 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 EXAMPLE Here is the neighborhood: [X, X, X] [0, X, 0] [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+1+x)*(1/y+y)+1 mod 2; OddCA(f, 70); MATHEMATICA (* f = A246038 *) f=1; f=7; f=29; f=103; f=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&], {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. A246037. Cf. A246038. Sequence in context: A111217 A198341 A299338 * A186142 A188274 A255281 Adjacent sequences:  A246036 A246037 A246038 * A246040 A246041 A246042 KEYWORD nonn AUTHOR N. J. A. Sloane, Aug 21 2014 STATUS approved

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Last modified October 21 13:57 EDT 2019. Contains 328299 sequences. (Running on oeis4.)