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A271061 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 261", based on the 5-celled von Neumann neighborhood.

%I A271061 #20 Mar 14 2025 08:59:49
%S A271061 1,8,48,224,960,3968,16128,65024,261120,1046528,4190208,16769024,
%T A271061 67092480,268402688,1073676288,4294836224
%N A271061 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 261", based on the 5-celled von Neumann neighborhood.
%C A271061 Initialized with a single black (ON) cell at stage zero.
%C A271061 Similar to A211012.
%C A271061 It is conjectured that Rules 269, 277, 285, 293, 301, 309, 317, 325, 333, 341, 349, 357, 365, 373, 381, 413, 445, 477, 509, 645, 653, 661, 669, 677, 685, 693, 701, 709, 717, 725, 733, 741, 749, 757 and 765 also produces this sequence.  It would be nice to have a proof.
%D A271061 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A271061 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 A271061 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A271061 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A271061 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A271061 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A271061 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A271061 Conjecture: a(n) = 4*4^n - 4*2^n, n>0. - _Lars Blomberg_, Jun 09 2016
%F A271061 Conjectures from _Colin Barker_, Dec 01 2016: (Start)
%F A271061 a(n) = 6*a(n-1) - 8*a(n-2) for n>2.
%F A271061 G.f.: (1 + 2*x + 8*x^2) / ((1 - 2*x) * (1 - 4*x)). (End)
%F A271061 Conjectured e.g.f.: (1 - 2*exp(2*x))^2. - _Stefano Spezia_, Mar 12 2025
%t A271061 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A271061 code=261; stages=128;
%t A271061 rule=IntegerDigits[code,2,10];
%t A271061 g=2*stages+1; (* Maximum size of grid *)
%t A271061 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A271061 ca=a;
%t A271061 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A271061 PrependTo[ca,a];
%t A271061 (* Trim full grid to reflect growth by one cell at each stage *)
%t A271061 k=(Length[ca[[1]]]+1)/2;
%t A271061 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 A271061 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A271061 Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)
%Y A271061 Cf. A211012, A271060.
%K A271061 nonn,more
%O A271061 0,2
%A A271061 _Robert Price_, Mar 29 2016
%E A271061 a(8)-a(15) from _Lars Blomberg_, Jun 09 2016