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%I #17 Jul 12 2017 06:20:28
%S 1,7,7,27,7,49,27,113,7,49,49,189,27,189,113,447,7,49,49,189,49,343,
%T 189,791,27,189,189,729,113,791,447,1743,7,49,49,189,49,343,189,791,
%U 49,343,343,1323,189,1323,791,3129,27,189,189,729,189,1323,729,3051,113,791,791,3051,447,3129,1743,6789,7
%N Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y)-y/x-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 177 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
%H Alois P. Heinz, <a href="/A255277/b255277.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 A255278.
%e Here is the neighborhood f:
%e [0, 0, X]
%e [X, X, X]
%e [X, X, X]
%e which contains a(1) = 7 ON cells.
%t (* f = A255278 *) f[0]=1; f[1]=7; f[2]=27; f[3]=113; f[4]=447; f[5]=1743; f[6]=6789; f[n_] := f[n] = -12f[n-8] - 4f[n-7] + 6f[n-6] - 4f[n-5] - 11f[n-4] - 7f[n-3] + 6f[n-2] + 3f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 64}] (* _Jean-François Alcover_, Jul 12 2017 *)
%Y Cf. A255278.
%K nonn
%O 0,2
%A _N. J. A. Sloane_ and _Doron Zeilberger_, Feb 19 2015
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