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A269696 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 6", based on the 5-celled von Neumann neighborhood.

%I A269696 #27 Feb 16 2025 08:33:30
%S A269696 1,5,20,80,320,1280,5120,20480,81920,327680,1310720,5242880,20971520,
%T A269696 83886080,335544320,1342177280
%N A269696 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 6", based on the 5-celled von Neumann neighborhood.
%C A269696 Initialized with a single black (ON) cell at stage zero.
%C A269696 Rules 38, 70, 102, 134, 166, 198 and 230 also generate this sequence.
%C A269696 Apparently a duplicate of A003947. - _R. J. Mathar_, Mar 09 2016
%D A269696 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A269696 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 A269696 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A269696 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A269696 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A269696 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A269696 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A269696 Conjectures from _Colin Barker_, Mar 08 2016: (Start)
%F A269696 a(n) = 5*4^(n-1) for n>0.
%F A269696 a(n) = 4*a(n-1) for n>1.
%F A269696 G.f.: (1+x) / (1-4*x).
%F A269696 (End)
%t A269696 rule=6; stages=300;
%t A269696 ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)
%t A269696 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A269696 Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)
%Y A269696 Cf. A269695.
%K A269696 nonn,easy
%O A269696 0,2
%A A269696 _Robert Price_, Mar 03 2016
%E A269696 a(9)-a(15) from _Lars Blomberg_, Apr 12 2016