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A247640 Number of ON cells after n generations of "Odd-Rule" cellular automaton on hexagonal lattice based on 6-celled neighborhood. 6

%I #22 Sep 07 2016 00:24:20

%S 1,6,6,24,6,36,24,96,6,36,36,144,24,144,96,384,6,36,36,144,36,216,144,

%T 576,24,144,144,576,96,576,384,1536,6,36,36,144,36,216,144,576,36,216,

%U 216,864,144,864,576,2304,24,144,144,576,144,864

%N Number of ON cells after n generations of "Odd-Rule" cellular automaton on hexagonal lattice based on 6-celled neighborhood.

%C The neighborhood of a cell consists of the six surrounding cells (but not the cell itself). A cell is ON at generation n iff an odd number of its neighbors were ON at the previous generation. We start with one ON cell.

%C This is the Run Length Transform of the sequence 1, 6, 24, 96, 384, 1536, 6144, 24576, ... (almost certainly A164908, or 1 followed by A002023).

%C It appears that this is also the sequence corresponding to the odd-rule cellular automaton defined by OddRule 356 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - _N. J. A. Sloane_, Feb 26 2015

%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 a(n) = number of terms in expansion of f^n mod 2, where f = 1/x+x+1/y+y+1/(x*y)+x*y (mod 2);

%p C := f->`if`(type(f,`+`),nops(f),1);

%p f := 1/x+x+1/y+y+1/(x*y)+x*y;

%p g := n->expand(f^n) mod 2;

%p [seq(C(g(n)),n=0..100)];

%t A247640[n_] := Total[CellularAutomaton[{42, {2, {{1, 1, 0}, {1, 0, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 2]; Array[A247640, 54, 0] (* _JungHwan Min_, Sep 06 2016 *)

%t A247640L[n_] := Total[#, 2] & /@ CellularAutomaton[{42, {2, {{1, 1, 0}, {1, 0, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, n]; A247640L[53] (* _JungHwan Min_, Sep 06 2016 *)

%Y Cf. A164908, A001023, A071053, A160239, A247666.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Sep 22 2014

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)