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A283507 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 641", based on the 5-celled von Neumann neighborhood.

%I A283507 #10 Feb 16 2025 08:33:42
%S A283507 1,0,5,3,23,15,95,63,383,255,1535,1023,6143,4095,24575,16383,98303,
%T A283507 65535,393215,262143,1572863,1048575,6291455,4194303,25165823,
%U A283507 16777215,100663295,67108863,402653183,268435455,1610612735,1073741823,6442450943,4294967295
%N A283507 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 641", based on the 5-celled von Neumann neighborhood.
%C A283507 Initialized with a single black (ON) cell at stage zero.
%D A283507 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A283507 Robert Price, <a href="/A283507/b283507.txt">Table of n, a(n) for n = 0..126</a>
%H A283507 Robert Price, <a href="/A283507/a283507.tmp.txt">Diagrams of first 20 stages</a>
%H A283507 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 A283507 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A283507 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A283507 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A283507 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A283507 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A283507 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A283507 Conjectures from _Colin Barker_, Mar 10 2017: (Start)
%F A283507 G.f.: (1 - x + x^2 + 2*x^3) / ((1 - x)*(1 - 2*x)*(1 + 2*x)).
%F A283507 a(n) = (-2 + (-2)^n + 2^(1+n)) / 2 for n>0.
%F A283507 a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n>3.
%F A283507 (End)
%t A283507 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A283507 code = 641; stages = 128;
%t A283507 rule = IntegerDigits[code, 2, 10];
%t A283507 g = 2 * stages + 1; (* Maximum size of grid *)
%t A283507 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A283507 ca = a;
%t A283507 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A283507 PrependTo[ca, a];
%t A283507 (* Trim full grid to reflect growth by one cell at each stage *)
%t A283507 k = (Length[ca[[1]]] + 1)/2;
%t A283507 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 A283507 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i ,1, stages - 1}]
%Y A283507 Cf. A283504, A283505, A283506.
%K A283507 nonn,easy
%O A283507 0,3
%A A283507 _Robert Price_, Mar 09 2017