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A269712 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 20", based on the 5-celled von Neumann neighborhood.

%I A269712 #21 Feb 16 2025 08:33:30
%S A269712 1,4,12,28,60,124,252,508,1020,2044,4092,8188,16380,32764,65532,131068
%N A269712 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 20", based on the 5-celled von Neumann neighborhood.
%C A269712 Initialized with a single black (ON) cell at stage zero.
%C A269712 Rules 28, 52, 60, 148, 156, 180, 188, 532, 540, 564, 572, 660, 668, 692  and 700 also generate this sequence.
%C A269712 Apparently a duplicate of A173033. - _R. J. Mathar_, Mar 09 2016
%D A269712 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A269712 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 A269712 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A269712 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A269712 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A269712 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A269712 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A269712 Conjectures from _Colin Barker_, Mar 08 2016: (Start)
%F A269712 a(n) = 4*(2^n-1) =A028399(n+2) for n>0.
%F A269712 a(n) = 3*a(n-1)-2*a(n-2) for n>2.
%F A269712 G.f.: (1+x+2*x^2) / ((1-x)*(1-2*x)).
%F A269712 (End)
%t A269712 rule=20; stages=300;
%t A269712 ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)
%t A269712 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A269712 Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)
%Y A269712 Cf. A269711.
%K A269712 nonn,more
%O A269712 0,2
%A A269712 _Robert Price_, Mar 04 2016
%E A269712 a(9)-a(15) from _Lars Blomberg_, Apr 15 2016