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A282124 Decimal 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 430", based on the 5-celled von Neumann neighborhood.

%I A282124 #20 Feb 16 2025 08:33:40
%S A282124 1,3,3,15,11,63,43,255,171,1023,683,4095,2731,16383,10923,65535,43691,
%T A282124 262143,174763,1048575,699051,4194303,2796203,16777215,11184811,
%U A282124 67108863,44739243,268435455,178956971,1073741823,715827883,4294967295,2863311531,17179869183
%N A282124 Decimal 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 430", based on the 5-celled von Neumann neighborhood.
%C A282124 Initialized with a single black (ON) cell at stage zero.
%D A282124 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A282124 Robert Price, <a href="/A282124/b282124.txt">Table of n, a(n) for n = 0..126</a>
%H A282124 Robert Price, <a href="/A282124/a282124.tmp.txt">Diagrams of first 20 stages</a>
%H A282124 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 A282124 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A282124 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A282124 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A282124 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A282124 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A282124 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A282124 Conjectures from _Colin Barker_, Feb 07 2017: (Start)
%F A282124 a(n) = (-1 + 2*(-1)^n - (-1)^n*2^(1+n) + 2^(2+n)) / 3.
%F A282124 a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
%F A282124 G.f.: (1 + 3*x - 2*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
%F A282124 (End)
%F A282124 Conjectures from _Paul Curtz_, Jun 10 2019: (Start)
%F A282124 a(n) = A001045(n+1)*(period 2: repeat[1, 3]).
%F A282124 a(n+4) = a(n) + 10*A081631(n).
%F A282124 a(2*n+1) = 2^(2*n+2) -1.
%F A282124 a(n+2) = a(n) + A098646(n+1).
%F A282124 (End)
%t A282124 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A282124 code = 430; stages = 128;
%t A282124 rule = IntegerDigits[code, 2, 10];
%t A282124 g = 2 * stages + 1; (* Maximum size of grid *)
%t A282124 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A282124 ca = a;
%t A282124 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A282124 PrependTo[ca, a];
%t A282124 (* Trim full grid to reflect growth by one cell at each stage *)
%t A282124 k = (Length[ca[[1]]] + 1)/2;
%t A282124 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 A282124 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i ,1, stages - 1}]
%Y A282124 Cf. A282121, A282122, A282123.
%Y A282124 Cf. A001045, A010684, A024036, A081631, A098646.
%K A282124 nonn,easy
%O A282124 0,2
%A A282124 _Robert Price_, Feb 06 2017