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A283508 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

%I A283508 #15 Feb 16 2025 08:33:42
%S A283508 1,10,101,1011,10111,101111,1011111,10111111,101111111,1011111111,
%T A283508 10111111111,101111111111,1011111111111,10111111111111,
%U A283508 101111111111111,1011111111111111,10111111111111111,101111111111111111,1011111111111111111,10111111111111111111
%N A283508 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.
%C A283508 Initialized with a single black (ON) cell at stage zero.
%C A283508 As far as the b-files reach (125 terms) this is the same as A267623. - _R. J. Mathar_, Mar 17 2017
%D A283508 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A283508 Robert Price, <a href="/A283508/b283508.txt">Table of n, a(n) for n = 0..126</a>
%H A283508 Robert Price, <a href="/A283508/a283508.tmp.txt">Diagrams of first 20 stages</a>
%H A283508 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 A283508 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A283508 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A283508 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A283508 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A283508 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A283508 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A283508 Conjectures from _Colin Barker_, Mar 10 2017: (Start)
%F A283508 G.f.: (1 - x + x^2) / ((1 - x)*(1 - 10*x)).
%F A283508 a(n) = (91*10^n - 10) / 90 for n>0.
%F A283508 a(n) = 11*a(n-1) - 10*a(n-2) for n>2.
%F A283508 (End)
%F A283508 Equivalent conjecture: a(n) = A267623(n). - _R. J. Mathar_, Mar 17 2017
%t A283508 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A283508 code = 643; stages = 128;
%t A283508 rule = IntegerDigits[code, 2, 10];
%t A283508 g = 2 * stages + 1; (* Maximum size of grid *)
%t A283508 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A283508 ca = a;
%t A283508 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A283508 PrependTo[ca, a];
%t A283508 (* Trim full grid to reflect growth by one cell at each stage *)
%t A283508 k = (Length[ca[[1]]] + 1)/2;
%t A283508 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 A283508 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A283508 Cf. A267623, A036563, A153893.
%K A283508 nonn,easy
%O A283508 0,2
%A A283508 _Robert Price_, Mar 09 2017