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%I #16 Jul 12 2017 06:05:43
%S 1,8,8,32,8,64,32,140,8,64,64,256,32,256,140,580,8,64,64,256,64,512,
%T 256,1120,32,256,256,1024,140,1120,580,2300,8,64,64,256,64,512,256,
%U 1120,64,512,512,2048,256,2048,1120,4640,32,256,256,1024,256
%N Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y)-y.
%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 577 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
%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 A253105.
%e Here is the neighborhood f:
%e [X, 0, X]
%e [X, X, X]
%e [X, X, X]
%e which contains a(1) = 8 ON cells.
%t (* f = A253105 *) f[0]=1; f[1]=8; f[2]=32; f[3]=140; f[4]=580; f[5]=2300; f[n_] := f[n] = -8f[n-7] + 8f[n-6] + 12f[n-5] - 29f[n-4] + 6f[n-3] + 4f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 52}] (* _Jean-François Alcover_, Jul 12 2017 *)
%Y Cf. A253105.
%K nonn
%O 0,2
%A _N. J. A. Sloane_ and _Doron Zeilberger_, Feb 19 2015
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