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A284351 Binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood.

%I A284351 #12 Feb 16 2025 08:33:43
%S A284351 1,1,111,1101,11111,111101,1111111,11111101,111111111,1111111101,
%T A284351 11111111111,111111111101,1111111111111,11111111111101,
%U A284351 111111111111111,1111111111111101,11111111111111111,111111111111111101,1111111111111111111,11111111111111111101
%N A284351 Binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood.
%C A284351 Initialized with a single black (ON) cell at stage zero.
%D A284351 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A284351 Robert Price, <a href="/A284351/b284351.txt">Table of n, a(n) for n = 0..126</a>
%H A284351 Robert Price, <a href="/A284351/a284351.tmp.txt">Diagrams of first 20 stages</a>
%H A284351 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 A284351 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A284351 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A284351 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A284351 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A284351 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A284351 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A284351 Conjectures from _Colin Barker_, Mar 26 2017: (Start)
%F A284351 G.f.: (1 - 9*x + 100*x^2) / ((1 - x)*(1 + x)*(1 - 10*x)).
%F A284351 a(n) = (10^(n+1) - 1)/9 for n even.
%F A284351 a(n) = (10^(n+1) - 91)/9 for n odd.
%F A284351 a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n>2.
%F A284351 (End)
%t A284351 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A284351 code = 899; stages = 128;
%t A284351 rule = IntegerDigits[code, 2, 10];
%t A284351 g = 2 * stages + 1; (* Maximum size of grid *)
%t A284351 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A284351 ca = a;
%t A284351 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A284351 PrependTo[ca, a];
%t A284351 (* Trim full grid to reflect growth by one cell at each stage *)
%t A284351 k = (Length[ca[[1]]] + 1)/2;
%t A284351 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 A284351 Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 10], {i, 1, stages - 1}]
%Y A284351 Cf. A284352, A284353, A284354.
%K A284351 nonn,easy
%O A284351 0,3
%A A284351 _Robert Price_, Mar 25 2017