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A283353 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 619", based on the 5-celled von Neumann neighborhood.

%I A283353 #13 Feb 16 2025 08:33:42
%S A283353 1,2,4,14,28,62,124,254,508,1022,2044,4094,8188,16382,32764,65534,
%T A283353 131068,262142,524284,1048574,2097148,4194302,8388604,16777214,
%U A283353 33554428,67108862,134217724,268435454,536870908,1073741822,2147483644,4294967294,8589934588
%N A283353 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 619", based on the 5-celled von Neumann neighborhood.
%C A283353 Initialized with a single black (ON) cell at stage zero.
%D A283353 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A283353 Robert Price, <a href="/A283353/b283353.txt">Table of n, a(n) for n = 0..126</a>
%H A283353 Robert Price, <a href="/A283353/a283353.tmp.txt">Diagrams of first 20 stages</a>
%H A283353 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 A283353 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A283353 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A283353 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A283353 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A283353 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A283353 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A283353 Conjectures from _Colin Barker_, Mar 06 2017: (Start)
%F A283353 G.f.: (1 + 2*x)*(1 - 2*x + 3*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
%F A283353 a(n) = 2^(n + 1) - 4 for n>0 and even.
%F A283353 a(n) = 2^(n + 1) - 2 for n odd.
%F A283353 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>3.
%F A283353 (End)
%t A283353 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A283353 code = 619; stages = 128;
%t A283353 rule = IntegerDigits[code, 2, 10];
%t A283353 g = 2 * stages + 1; (* Maximum size of grid *)
%t A283353 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A283353 ca = a;
%t A283353 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A283353 PrependTo[ca, a];
%t A283353 (* Trim full grid to reflect growth by one cell at each stage *)
%t A283353 k = (Length[ca[[1]]] + 1)/2;
%t A283353 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 A283353 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i ,1, stages - 1}]
%Y A283353 Cf. A283351, A283352.
%K A283353 nonn,easy
%O A283353 0,2
%A A283353 _Robert Price_, Mar 05 2017