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A282577 Decimal 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 553", based on the 5-celled von Neumann neighborhood.

%I A282577 #20 Feb 16 2025 08:33:41
%S A282577 1,0,1,0,5,0,21,0,85,0,341,0,1365,0,5461,0,21845,0,87381,0,349525,0,
%T A282577 1398101,0,5592405,0,22369621,0,89478485,0,357913941,0,1431655765,0,
%U A282577 5726623061,0,22906492245,0,91625968981,0,366503875925,0,1466015503701,0
%N A282577 Decimal 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 553", based on the 5-celled von Neumann neighborhood.
%C A282577 Initialized with a single black (ON) cell at stage zero.
%D A282577 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A282577 Robert Price, <a href="/A282577/b282577.txt">Table of n, a(n) for n = 0..126</a>
%H A282577 Robert Price, <a href="/A282577/a282577.tmp.txt">Diagrams of first 20 stages</a>
%H A282577 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 A282577 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A282577 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A282577 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A282577 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A282577 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A282577 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A282577 Conjectures from _Colin Barker_, Feb 27 2017: (Start)
%F A282577 a(n) = (2^n - 1) / 3 for n>1 and even.
%F A282577 a(n) = 0 for n>1 and odd.
%F A282577 a(n) = 5*a(n-2) - 4*a(n-4) for n>4.
%F A282577 G.f.: (1 - 2*x^2)^2 / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
%F A282577 (End)
%t A282577 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A282577 code = 553; stages = 128;
%t A282577 rule = IntegerDigits[code, 2, 10];
%t A282577 g = 2 * stages + 1; (* Maximum size of grid *)
%t A282577 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A282577 ca = a;
%t A282577 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A282577 PrependTo[ca, a];
%t A282577 (* Trim full grid to reflect growth by one cell at each stage *)
%t A282577 k = (Length[ca[[1]]] + 1)/2;
%t A282577 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 A282577 Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 2], {i, 1, stages - 1}]
%Y A282577 Cf. A282088, A282142, A282579.
%K A282577 nonn,easy
%O A282577 0,5
%A A282577 _Robert Price_, Feb 27 2017