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%I #23 Jul 12 2017 05:24:27
%S 1,6,6,24,6,36,24,96,6,36,36,144,24,144,96,372,6,36,36,144,36,216,144,
%T 576,24,144,144,576,96,576,372,1416,6,36,36,144,36,216,144,576,36,216,
%U 216,864,144,864,576,2232,24,144,144,576,144,864,576,2304,96,576,576,2304,372,2232,1416,5340
%N Number of odd terms in f^n, where f = 1/(x*y)+1/x+1/x*y+1/y+x+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 347 (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 A253101.
%e Here is the neighborhood f:
%e [X, 0, X]
%e [X, 0, X]
%e [X, X, 0]
%e which contains a(1) = 6 ON cells.
%p C:=f->subs({x=1, y=1}, f);
%p # Find number of ON cells in CA for generations 0 thru M defined by rule
%p # that cell is ON iff number of ON cells in nbd at time n-1 was odd
%p # where nbd is defined by a polynomial or Laurent series f(x, y).
%p OddCA:=proc(f, M) global C; local n, a, i, f2, p;
%p f2:=simplify(expand(f)) mod 2;
%p a:=[]; p:=1;
%p for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
%p lprint([seq(a[i], i=1..nops(a))]);
%p end;
%p f:=1/(x*y)+1/x+1/x*y+1/y+x+x*y;
%p OddCA(f, 130);
%t (* f = A253101 *) f[n_] := 2*(2-Sqrt[3])^n + 2*(2+Sqrt[3])^n - 2^n // Round; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* _Jean-François Alcover_, Jul 12 2017 *)
%Y Cf. A253101. Similar to but different from A247640.
%K nonn
%O 0,2
%A _N. J. A. Sloane_ and _Doron Zeilberger_, Feb 19 2015
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