A255279 - OEIS

%I #20 Jul 12 2017 07:43:07

%S 1,7,7,25,7,49,25,107,7,49,49,175,25,175,107,413,7,49,49,175,49,343,

%T 175,749,25,175,175,625,107,749,413,1615,7,49,49,175,49,343,175,749,

%U 49,343,343,1225,175,1225,749,2891,25,175,175,625,175,1225

%N Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y)-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 277 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

%H Alois P. Heinz, <a href="/A255279/b255279.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 A255280.

%e Here is the neighborhood f:

%e [0, X, 0]

%e [X, X, X]

%e [X, X, X]

%e which contains a(1) = 7 ON cells.

%t (* f = A255280 *) f[0]=1; f[1]=7; f[2]=25; f[3]=107; f[4]=413; f[5]=1615; f[n_] := f[n] = -16 f[n-7] - 16 f[n-6] + 16 f[n-4] - 9 f[n-3] - 3 f[n-2] + 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. A255280.

%K nonn

%O 0,2

%A _N. J. A. Sloane_ and _Doron Zeilberger_, Feb 19 2015