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A280411 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 261", based on the 5-celled von Neumann neighborhood.

%I A280411 #11 Feb 16 2025 08:33:39
%S A280411 1,1,100,111,10000,11111,1000000,1111111,100000000,111111111,
%T A280411 10000000000,11111111111,1000000000000,1111111111111,100000000000000,
%U A280411 111111111111111,10000000000000000,11111111111111111,1000000000000000000,1111111111111111111
%N A280411 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 261", based on the 5-celled von Neumann neighborhood.
%C A280411 Initialized with a single black (ON) cell at stage zero.
%D A280411 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A280411 Robert Price, <a href="/A280411/b280411.txt">Table of n, a(n) for n = 0..126</a>
%H A280411 Robert Price, <a href="/A280411/a280411.tmp.txt">Diagrams of first 20 stages</a>
%H A280411 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 A280411 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A280411 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A280411 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A280411 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A280411 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A280411 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A280411 Conjectures from _Colin Barker_, Jan 03 2017: (Start)
%F A280411 a(n) = 2^n*(4*(-5)^n + 5^(n + 1))/9 for n even.
%F A280411 a(n) = ((-5)^n*2^(n+2) + 2^n*5^(n+1) - 1)/9 for n odd.
%F A280411 a(n) = 101*a(n-2) - 100*a(n-4) for n>3.
%F A280411 G.f.: (1 + x - x^2 + 10*x^3) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
%F A280411 (End)
%t A280411 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A280411 code = 261; stages = 128;
%t A280411 rule = IntegerDigits[code, 2, 10];
%t A280411 g = 2 * stages + 1; (* Maximum size of grid *)
%t A280411 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A280411 ca = a;
%t A280411 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A280411 PrependTo[ca, a];
%t A280411 (* Trim full grid to reflect growth by one cell at each stage *)
%t A280411 k = (Length[ca[[1]]] + 1)/2;
%t A280411 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 A280411 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A280411 Cf. A280410, A280412, A051049.
%K A280411 nonn,easy
%O A280411 0,3
%A A280411 _Robert Price_, Jan 02 2017