|
%I #24 Jul 12 2017 05:30:07
%S 1,6,6,22,6,36,22,82,6,36,36,132,22,132,82,302,6,36,36,132,36,216,132,
%T 492,22,132,132,484,82,492,302,1106,6,36,36,132,36,216,132,492,36,216,
%U 216,792,132,792,492,1812,22,132,132,484,132,792,484,1804,82,492,492,1804,302,1812,1106,4066
%N Number of odd terms in f^n, where f = 1/x+1+x+x/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 175 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
%H Alois P. Heinz, <a href="/A253069/b253069.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 A253070.
%e Here is the neighborhood f:
%e [X, 0, X]
%e [X, X, X]
%e [0, 0, X]
%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+x/y+y/x+x*y;
%p OddCA(f, 130);
%t (* f = A253070 *) f[0]=1; f[1]=6; f[2]=22; f[3]=82; f[4]=302; f[5]=1106;f[6]=4066; f[n_] := f[n] = 8 f[n-4] + 8 f[n-3] + 3 f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[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, A253066.
%Y Cf. A253070.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jan 29 2015
|
