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A282295 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 451", based on the 5-celled von Neumann neighborhood.

%I A282295 #10 Jun 21 2025 13:26:06
%S A282295 1,1,111,0,11111,0,1111111,0,111111111,0,11111111111,0,1111111111111,
%T A282295 0,111111110011111,0,11111111000000111,11110000,1111111100110011011,
%U A282295 11110000,111111110000000000011,111111111000,11111111111100000001111,111111110000
%N A282295 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 451", based on the 5-celled von Neumann neighborhood.
%C A282295 Initialized with a single black (ON) cell at stage zero.
%D A282295 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A282295 Robert Price, <a href="/A282295/b282295.txt">Table of n, a(n) for n = 0..126</a>
%H A282295 Robert Price, <a href="/A282295/a282295.tmp.txt">Diagrams of first 20 stages</a>
%H A282295 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 A282295 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A282295 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A282295 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A282295 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A282295 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A282295 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A282295 Conjecture: a(n) = A279720(n) for n>=2. - _R. J. Mathar_, Jun 21 2025
%t A282295 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A282295 code = 451; stages = 128;
%t A282295 rule = IntegerDigits[code, 2, 10];
%t A282295 g = 2 * stages + 1; (* Maximum size of grid *)
%t A282295 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A282295 ca = a;
%t A282295 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A282295 PrependTo[ca, a];
%t A282295 (* Trim full grid to reflect growth by one cell at each stage *)
%t A282295 k = (Length[ca[[1]]] + 1)/2;
%t A282295 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 A282295 Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 10], {i, 1, stages - 1}]
%Y A282295 Cf. A282297, A282298, A282299.
%K A282295 nonn,easy
%O A282295 0,3
%A A282295 _Robert Price_, Feb 11 2017