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A247666 Number of ON cells after n generations of "Odd-Rule" cellular automaton on hexagonal lattice based on 7-celled neighborhood.

%I A247666 #53 Dec 31 2020 11:11:15
%S A247666 1,7,7,25,7,49,25,103,7,49,49,175,25,175,103,409,7,49,49,175,49,343,
%T A247666 175,721,25,175,175,625,103,721,409,1639,7,49,49,175,49,343,175,721,
%U A247666 49,343,343,1225,175,1225,721,2863,25,175,175,625
%N A247666 Number of ON cells after n generations of "Odd-Rule" cellular automaton on hexagonal lattice based on 7-celled neighborhood.
%C A247666 The neighborhood of a cell consists of the cell itself together with its six surrounding cells. 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 A247666 This is the Run Length Transform of the sequence 1,7,25,103,409,1639,26215,... (almost certainly A102900).
%C A247666 This appears to be the same as the number of ON cells in a certain 2-D CA on the square grid in which the neighborhood of a cell is defined by f = 1/(x*y)+1/x+1/x*y+1/y+x/y+x+x*y, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation. Here is the neighborhood:
%C A247666 [X, 0, X]
%C A247666 [X, 0, X]
%C A247666 [X, X, X]
%C A247666 which contains a(1) = 7 ON cells.
%C A247666 This is the odd-rule cellular automaton defined by OddRule 557 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
%C A247666 Furthermore, this is also the number of ON cells in the 2-D CA on the square grid in which the neighborhood of a cell is defined by f = 1/(x*y)+1/x+1/y+1+y+x+x*y, with the same rule. Here is the neighborhood:
%C A247666 [0, X, X]
%C A247666 [X, X, X]
%C A247666 [X, X, 0]
%C A247666 - _N. J. A. Sloane_, Feb 19 2015
%C A247666 This is the odd-rule cellular automaton defined by OddRule 376 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
%C A247666 The partial sums are in A253767 in which the structure looks like an irregular stepped pyramid, apparently with a like-hexagonal base. - _Omar E. Pol_, Jan 29 2015
%H A247666 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 A247666 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 A247666 N. J. A. Sloane, <a href="/A247666/a247666.pdf">Illustrations of generations 0 to 4</a>
%H A247666 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 A247666 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 A247666 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A247666 a(n) = number of terms in expansion of f^n mod 2, where f = 1+1/x+x+1/y+y+1/(x*y)+x*y (mod 2);
%e A247666 From _Omar E. Pol_, Jan 29 2015: (Start)
%e A247666 May be arranged into blocks of sizes A011782:
%e A247666 1;
%e A247666 7;
%e A247666 7, 25;
%e A247666 7, 49, 25, 103;
%e A247666 7, 49, 49, 175, 25, 175, 103, 409;
%e A247666 7, 49, 49, 175, 49, 343, 175, 721, 25, 175, 175, 625, 103, 721, 409, 1639;
%e A247666 7, 49, 49, 175, 49, 343, 175, 721, 49, 343, 343, 1225, 175, 1225, 721, 2863, 25, 175, 175, 625, ...
%e A247666 It appears that right border gives A102900 without repetitions, see Comments section. [This is just a restatement of the fact that this sequence is the run length transform of what is presumably A102900. - _N. J. A. Sloane_, Feb 06 2015]
%e A247666 (End)
%e A247666 From _Omar E. Pol_, Mar 19 2015: (Start)
%e A247666 Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below:
%e A247666 1;
%e A247666 ..
%e A247666 7;
%e A247666 ..
%e A247666 7;
%e A247666 25;
%e A247666 .........
%e A247666 7,    49;
%e A247666 25;
%e A247666 103;
%e A247666 ...................
%e A247666 7,    49,  49, 175;
%e A247666 25,  175;
%e A247666 103;
%e A247666 409;
%e A247666 ......................................
%e A247666 7,    49,  49, 175, 49, 343, 175, 721;
%e A247666 25,  175, 175, 625;
%e A247666 103, 721;
%e A247666 409;
%e A247666 1639;
%e A247666 ...
%e A247666 Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k).
%e A247666 (End)
%p A247666 C := f->`if`(type(f,`+`),nops(f),1);
%p A247666 f := 1+1/x+x+1/y+y+1/(x*y)+x*y;
%p A247666 g := n->expand(f^n) mod 2;
%p A247666 [seq(C(g(n)),n=0..100)];
%t A247666 A247666[n_] := Total[CellularAutomaton[{170, {2, {{1, 1, 0}, {1, 1, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 2]; Array[A247666, 52, 0] (* _JungHwan Min_, Sep 01 2016 *)
%t A247666 A247666L[n_] := Total[#, 2] & /@ CellularAutomaton[{170, {2, {{1, 1, 0}, {1, 1, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, n]; A247666L[51] (* _JungHwan Min_, Sep 01 2016 *)
%Y A247666 Cf. A102900, A071053, A160239, A247640.
%K A247666 nonn
%O A247666 0,2
%A A247666 _N. J. A. Sloane_, Sep 22 2014