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A270218 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 129", based on the 5-celled von Neumann neighborhood.

%I A270218 #31 Feb 16 2025 08:33:31
%S A270218 1,4,28,140,620,2604,10668,43180,173740,697004,2792108,11176620,
%T A270218 44722860,178924204,715762348,2863180460
%N A270218 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 129", based on the 5-celled von Neumann neighborhood.
%C A270218 Initialized with a single black (ON) cell at stage zero.
%C A270218 It appears Rules 385, 425, 465 and 553 also generate this sequence. - _Lars Blomberg_, Apr 30 2016 (It would be nice to have a proof! - _N. J. A. Sloane_, May 09 2016)
%D A270218 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A270218 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 A270218 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A270218 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A270218 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A270218 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A270218 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A270218 Conjectures from _Colin Barker_, Mar 13 2016: (Start)
%F A270218 a(n) = 4*(1-3*2^n+2^(1+2*n))/3.
%F A270218 a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.
%F A270218 G.f.: (1-3*x+14*x^2-8*x^3) / ((1-x)*(1-2*x)*(1-4*x)).
%F A270218 (End)
%F A270218 a(n) = 4*A006095(n+1) (conjectured). - _Michal Stajszczak_, May 20 2020
%t A270218 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A270218 code=129; stages=128;
%t A270218 rule=IntegerDigits[code,2,10];
%t A270218 g=2*stages+1; (* Maximum size of grid *)
%t A270218 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A270218 ca=a;
%t A270218 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A270218 PrependTo[ca,a];
%t A270218 (* Trim full grid to reflect growth by one cell at each stage *)
%t A270218 k=(Length[ca[[1]]]+1)/2;
%t A270218 ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
%t A270218 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A270218 Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)
%Y A270218 Cf. A270217.
%K A270218 nonn,more
%O A270218 0,2
%A A270218 _Robert Price_, Mar 13 2016
%E A270218 a(8)-a(15) from _Lars Blomberg_, Apr 30 2016