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A282414 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 467", based on the 5-celled von Neumann neighborhood.

%I A282414 #16 Jun 21 2025 13:18:26
%S A282414 1,2,7,0,31,0,127,0,511,0,2047,0,8191,0,32767,0,131071,0,524287,0,
%T A282414 2097151,0,8388607,0,33554431,0,134217727,0,536870911,0,2147483647,0,
%U A282414 8589934591,0,34359738367,0,137438953471,0,549755813887,0,2199023255551,0
%N A282414 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 467", based on the 5-celled von Neumann neighborhood.
%C A282414 Initialized with a single black (ON) cell at stage zero.
%C A282414 For n != 1, is a(n) = A279872(n)? - _Bruno Berselli_, Feb 15 2017
%D A282414 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A282414 Robert Price, <a href="/A282414/b282414.txt">Table of n, a(n) for n = 0..126</a>
%H A282414 Robert Price, <a href="/A282414/a282414.tmp.txt">Diagrams of first 20 stages</a>
%H A282414 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 A282414 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A282414 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A282414 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A282414 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A282414 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A282414 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A282414 Conjectures from _Colin Barker_, Feb 15 2017: (Start)
%F A282414 a(n) = 2^(n+1)-1 for n>1 and even.
%F A282414 a(n) = 0 for n>1 and odd.
%F A282414 a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
%F A282414 G.f.: (1 + 2*x + 2*x^2 - 10*x^3 + 8*x^5) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
%F A282414 (End)
%F A282414 Equivalent conjecture: a(n) = A282413(n) for n>=2. - _R. J. Mathar_, Jun 21 2025
%t A282414 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A282414 code = 467; stages = 128;
%t A282414 rule = IntegerDigits[code, 2, 10];
%t A282414 g = 2 * stages + 1; (* Maximum size of grid *)
%t A282414 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A282414 ca = a;
%t A282414 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A282414 PrependTo[ca, a];
%t A282414 (* Trim full grid to reflect growth by one cell at each stage *)
%t A282414 k = (Length[ca[[1]]] + 1)/2;
%t A282414 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 A282414 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i ,1, stages - 1}]
%Y A282414 Cf. A282411, A282412, A282413, A279872.
%K A282414 nonn,easy
%O A282414 0,2
%A A282414 _Robert Price_, Feb 14 2017