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A281146 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 515", based on the 5-celled von Neumann neighborhood.

%I A281146 #33 Feb 16 2025 08:33:39
%S A281146 1,10,100,1000,10100,101000,1010100,10101000,101010100,1010101000,
%T A281146 10101010100,101010101000,1010101010100,10101010101000,
%U A281146 101010101010100,1010101010101000,10101010101010100,101010101010101000,1010101010101010100,10101010101010101000
%N A281146 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 515", based on the 5-celled von Neumann neighborhood.
%C A281146 Initialized with a single black (ON) cell at stage zero.
%D A281146 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A281146 Robert Price, <a href="/A281146/b281146.txt">Table of n, a(n) for n = 0..126</a>
%H A281146 Robert Price, <a href="/A281146/a281146.tmp.txt">Diagrams of first 20 stages</a>
%H A281146 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 A281146 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A281146 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A281146 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A281146 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A281146 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A281146 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A281146 Conjectures from _Colin Barker_, Feb 22 2017: (Start)
%F A281146 a(n) = 100*(10^n - 1) / 99 for n>1 and even.
%F A281146 a(n) = 100*(10^n - 10) / 99 for n>1 and odd.
%F A281146 a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n>2.
%F A281146 G.f.: (1 - x^2 + 100*x^4) / ((1 - x)*(1 + x)*(1 - 10*x)).
%F A281146 (End)
%t A281146 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A281146 code = 515; stages = 128;
%t A281146 rule = IntegerDigits[code, 2, 10];
%t A281146 g = 2 * stages + 1; (* Maximum size of grid *)
%t A281146 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A281146 ca = a;
%t A281146 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A281146 PrependTo[ca, a];
%t A281146 (* Trim full grid to reflect growth by one cell at each stage *)
%t A281146 k = (Length[ca[[1]]] + 1)/2;
%t A281146 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 A281146 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A281146 Cf. A052992, A153772, A279053.
%K A281146 nonn,easy
%O A281146 0,2
%A A281146 _Robert Price_, Feb 22 2017