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A273335 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 657", based on the 5-celled von Neumann neighborhood.

%I A273335 #15 Mar 27 2025 11:21:38
%S A273335 1,4,48,224,960,3968,16128,65024,261120,1046528,4190208,16769024,
%T A273335 67092480,268402688,1073676288,4294836224
%N A273335 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 657", based on the 5-celled von Neumann neighborhood.
%C A273335 Initialized with a single black (ON) cell at stage zero.
%C A273335 Conjecture: Rule 665 also generates this sequence. - _Lars Blomberg_, Jul 18 2016
%C A273335 Seems to differ from A271061 only at n=1. - _R. J. Mathar_, Mar 27 2025
%D A273335 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A273335 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 A273335 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A273335 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A273335 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A273335 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A273335 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A273335 Conjectures from _Colin Barker_, May 20 2016: (Start)
%F A273335 a(n) = 2^(n+2)*(2^n-1) for n>1.
%F A273335 a(n) = 6*a(n-1)-8*a(n-2) for n>3.
%F A273335 G.f.: (1-2*x+32*x^2-32*x^3) / ((1-2*x)*(1-4*x)).
%F A273335 (End)
%t A273335 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A273335 code=657; stages=128;
%t A273335 rule=IntegerDigits[code,2,10];
%t A273335 g=2*stages+1; (* Maximum size of grid *)
%t A273335 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A273335 ca=a;
%t A273335 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A273335 PrependTo[ca,a];
%t A273335 (* Trim full grid to reflect growth by one cell at each stage *)
%t A273335 k=(Length[ca[[1]]]+1)/2;
%t A273335 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 A273335 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A273335 Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)
%Y A273335 Cf. A273334.
%K A273335 nonn,more
%O A273335 0,2
%A A273335 _Robert Price_, May 20 2016
%E A273335 a(8)-a(15) from _Lars Blomberg_, Jul 18 2016