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A269708 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.

%I A269708 #18 Feb 16 2025 08:33:30
%S A269708 1,5,20,76,292,1132,4420,17356,68452,270892,1074820,4273036,17013412,
%T A269708 67817452,270561220,1080119116
%N A269708 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.
%C A269708 Initialized with a single black (ON) cell at stage zero.
%C A269708 Rules 46, 142 and 174 also generate this sequence.
%D A269708 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A269708 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 A269708 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A269708 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A269708 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A269708 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A269708 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A269708 Conjectures from _Colin Barker_, Mar 08 2016: (Start)
%F A269708 a(n) = 4*3^(n-2)+4^n for n>1.
%F A269708 a(n) = 7*a(n-1)-12*a(n-2) for n>3.
%F A269708 G.f.: (1-2*x-3*x^2-4*x^3) / ((1-3*x)*(1-4*x)).
%F A269708 (End)
%t A269708 rule=14; stages=300;
%t A269708 ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)
%t A269708 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A269708 Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)
%Y A269708 Cf. A269707.
%K A269708 nonn,more
%O A269708 0,2
%A A269708 _Robert Price_, Mar 04 2016
%E A269708 a(9)-a(15) from _Lars Blomberg_, Apr 12 2016