A253066 - OEIS

%I #24 Jul 12 2017 03:20:18

%S 1,6,6,28,6,36,28,112,6,36,36,168,28,168,112,456,6,36,36,168,36,216,

%T 168,672,28,168,168,784,112,672,456,1816,6,36,36,168,36,216,168,672,

%U 36,216,216,1008,168,1008,672,2736,28,168,168,784,168,1008,784,3136,112,672,672,3136,456,2736,1816,7288

%N Number of odd terms in f^n, where f = 1/x+1+x+1/y+y/x+x*y.

%C This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.

%C This is the odd-rule cellular automaton defined by OddRule 275 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

%H Alois P. Heinz, <a href="/A253066/b253066.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 [math.CO], 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 [math.CO], 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 [math.CO], 2015.

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%F This is the Run Length Transform of A253068.

%e Here is the neighborhood f:

%e [X, 0, X]

%e [X, X, X]

%e [0, 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+1+x+1/y+y/x+x*y;

%p OddCA(f, 130);

%t (* f = A253068 *) f[0] = 1; f[n_] := ((-2)^n + 4^(n+2)-8)/9; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1 &], {n, 0, 63}] (* _Jean-François Alcover_, Jul 12 2017 *)

%Y Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035, A253064, A253065.

%Y Cf. A253068.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Jan 29 2015