cp's OEIS Frontend

A282088 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 553", based on the 5-celled von Neumann neighborhood.

%I A282088 #27 Feb 16 2025 08:33:40
%S A282088 1,0,1,0,101,0,10101,0,1010101,0,101010101,0,10101010101,0,
%T A282088 1010101010101,0,101010101010101,0,10101010101010101,0,
%U A282088 1010101010101010101,0,101010101010101010101,0,10101010101010101010101,0,1010101010101010101010101,0
%N A282088 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 553", based on the 5-celled von Neumann neighborhood.
%C A282088 Initialized with a single black (ON) cell at stage zero.
%D A282088 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A282088 Robert Price, <a href="/A282088/b282088.txt">Table of n, a(n) for n = 0..126</a>
%H A282088 Robert Price, <a href="/A282088/a282088.tmp.txt">Diagrams of first 20 stages</a>
%H A282088 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 A282088 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A282088 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A282088 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A282088 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A282088 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A282088 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A282088 Conjectures from _Colin Barker_, Feb 27 2017: (Start)
%F A282088 a(n) = 101*a(n-2) - 100*a(n-4) for n>4.
%F A282088 a(n) = (10^n - 1) / 99 for n>0 and even.
%F A282088 a(n) = 0 for n odd.
%F A282088 G.f.: (1 - 100*x^2 + 100*x^4) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
%F A282088 (End)
%t A282088 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A282088 code = 553; stages = 128;
%t A282088 rule = IntegerDigits[code, 2, 10];
%t A282088 g = 2 * stages + 1; (* Maximum size of grid *)
%t A282088 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A282088 ca = a;
%t A282088 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A282088 PrependTo[ca, a];
%t A282088 (* Trim full grid to reflect growth by one cell at each stage *)
%t A282088 k = (Length[ca[[1]]] + 1)/2;
%t A282088 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 A282088 Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 10], {i, 1, stages - 1}]
%Y A282088 Cf. A282142, A282577, A282579.
%K A282088 nonn,easy
%O A282088 0,5
%A A282088 _Robert Price_, Feb 27 2017