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A246034 Number of odd terms in f^n, where f = x^4*y^4 + x^4*y^3 + x^3*y^4 + x^4*y^2 + x^2*y^4 + x^4*y + x^3*y^2 + x^2*y^3 + x*y^4 + x^4 + x^2*y^2 + y^4 + x^3 + x^2*y + x*y^2 + y^3 + x^2 + y^2 + x + y + 1. 9

%I

%S 1,21,21,85,21,233,85,321,21,441,233,761,85,1137,321,1545,21,441,441,

%T 1785,233,2925,761,3589,85,1785,1137,3977,321,4549,1545,5909,21,441,

%U 441,1785,441,4893,1785,6741,233,4893,2925,9949,761,11301,3589,13181,85,1785,1785

%N Number of odd terms in f^n, where f = x^4*y^4 + x^4*y^3 + x^3*y^4 + x^4*y^2 + x^2*y^4 + x^4*y + x^3*y^2 + x^2*y^3 + x*y^4 + x^4 + x^2*y^2 + y^4 + x^3 + x^2*y + x*y^2 + y^3 + x^2 + y^2 + x + y + 1.

%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.

%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>

%e Here is the neighborhood:

%e [X, X, X, X, X]

%e [X, 0, X, 0, X]

%e [X, X, X, X, X]

%e [X, 0, X, 0, X]

%e [X, X, X, X, X]

%e which contains a(1) = 21 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:=x^4*y^4+x^4*y^3+x^3*y^4+x^4*y^2+x^2*y^4+x^4*y+x^3*y^2+x^2*y^3+x*y^4+x^4+

%p x^2*y^2+y^4+x^3+x^2*y+x*y^2+y^3+x^2+y^2+x+y+1;

%p OddCA(f, 100);

%t f = x^4*y^4 + x^4*y^3 + x^3*y^4 + x^4*y^2 + x^2*y^4 + x^4*y + x^3*y^2 + x^2*y^3 + x*y^4 + x^4 + x^2*y^2 + y^4 + x^3 + x^2*y + x*y^2 + y^3 + x^2 + y^2 + x + y + 1;

%t a[0] = 1; a[n_] := Count[List @@ Expand[f^n] /. {x -> 1, y -> 1}, _?OddQ];

%t Table[a[n], {n, 0, 50}] (* _Jean-Fran├žois Alcover_, Dec 11 2017 *)

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

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Aug 20 2014

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