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%I A280371 #11 Jan 02 2017 02:58:23
%S A280371 1,10,101,1000,10000,101000,1010100,10000010,100000001,1010000000,
%T A280371 10101000000,100000100000,1000000000000,10100000000000,
%U A280371 101010000000000,1000001000000000,10000000000000000,101000000000000000,1010100000000000000,10000010000000000000
%N A280371 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 260", based on the 5-celled von Neumann neighborhood.
%C A280371 Initialized with a single black (ON) cell at stage zero.
%D A280371 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A280371 Robert Price, <a href="/A280371/b280371.txt">Table of n, a(n) for n = 0..126</a>
%H A280371 Robert Price, <a href="/A280371/a280371.tmp.txt">Diagrams of first 20 stages</a>
%H A280371 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 A280371 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A280371 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A280371 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A280371 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A280371 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A280371 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A280371 Conjectures from _Colin Barker_, Jan 02 2017: (Start)
%F A280371 a(n) = 10000*a(n-4) for n>3.
%F A280371 G.f.: (1 + 10*x + 101*x^2 + 1000*x^3 + 1000*x^5 + 100*x^6 + 10*x^7 + x^8 - 10000*x^12) / ((1 - 10*x)*(1 + 10*x)*(1 + 100*x^2)).
%F A280371 (End)
%t A280371 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A280371 code = 260; stages = 128;
%t A280371 rule = IntegerDigits[code, 2, 10];
%t A280371 g = 2 * stages + 1; (* Maximum size of grid *)
%t A280371 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A280371 ca = a;
%t A280371 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A280371 PrependTo[ca, a];
%t A280371 (* Trim full grid to reflect growth by one cell at each stage *)
%t A280371 k = (Length[ca[[1]]] + 1)/2;
%t A280371 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 A280371 Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 10], {i, 1, stages - 1}]
%Y A280371 Cf. A280372, A280373, A280374.
%K A280371 nonn,easy
%O A280371 0,2
%A A280371 _Robert Price_, Jan 01 2017

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