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%I #16 Jul 12 2017 07:37:39
%S 1,7,7,31,7,49,31,127,7,49,49,217,31,217,127,511,7,49,49,217,49,343,
%T 217,889,31,217,217,961,127,889,511,2031,7,49,49,217,49,343,217,889,
%U 49,343,343,1519,217,1519,889,3577,31,217,217,961,217,1519
%N Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y)-y/x-x.
%C This is the number of ON cells in a certain two-dimensional cellular automaton in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there were an odd number of ON cells in the neighborhood at the previous generation.
%C This is the odd-rule cellular automaton defined by OddRule 367 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
%H Alois P. Heinz, <a href="/A255281/b255281.txt">Table of n, a(n) for n = 0..8191</a>
%H Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.01796">A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata</a>, arXiv:1503.01796, 2015; see also the <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/CAcount.html">Accompanying Maple Package</a>.
%H Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1503.04249">Odd-Rule Cellular Automata on the Square Grid</a>, arXiv:1503.04249, 2015.
%H 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: <a href="https://vimeo.com/119073818">Part 1</a>, <a href="https://vimeo.com/119073819">Part 2</a>
%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168, 2015
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F This is the Run Length Transform of A255282.
%e Here is the neighborhood f:
%e [0, X, X]
%e [X, X, 0]
%e [X, X, X]
%e which contains a(1) = 7 ON cells.
%t (* f = A255282 *) f[0]=1; f[1]=7; f[2]=31; f[3]=127; f[4]=511; f[5]=2031; f[6]=8043; f[7]=31735; f[8]=125063; f[n_] := f[n] = 10 f[n-10] + 22 f[n-9] - 11 f[n-8] + 31 f[n-7] - 24 f[n-6] + 3 f[n-5] + 18 f[n-4] - 21 f[n-3] + 5 f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 53}] (* _Jean-François Alcover_, Jul 12 2017 *)
%Y Cf. A255282.
%K nonn
%O 0,2
%A _N. J. A. Sloane_ and _Doron Zeilberger_, Feb 19 2015
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