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A280412 Decimal 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 261", based on the 5-celled von Neumann neighborhood.

%I A280412 #13 Feb 16 2025 08:33:39
%S A280412 1,2,1,14,1,62,1,254,1,1022,1,4094,1,16382,1,65534,1,262142,1,1048574,
%T A280412 1,4194302,1,16777214,1,67108862,1,268435454,1,1073741822,1,
%U A280412 4294967294,1,17179869182,1,68719476734,1,274877906942,1,1099511627774,1,4398046511102,1
%N A280412 Decimal 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 261", based on the 5-celled von Neumann neighborhood.
%C A280412 Initialized with a single black (ON) cell at stage zero.
%D A280412 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A280412 Robert Price, <a href="/A280412/b280412.txt">Table of n, a(n) for n = 0..126</a>
%H A280412 Robert Price, <a href="/A280412/a280412.tmp.txt">Diagrams of first 20 stages</a>
%H A280412 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 A280412 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A280412 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A280412 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A280412 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A280412 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A280412 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A280412 Conjectures from _Ilya Gutkovskiy_, Jan 02 2017: (Start)
%F A280412 G.f.: (1 + 2*x - 4*x^2 + 4*x^3)/((1 - 2*x)*(1 + 2*x)*(1 - x)*(1 + x)).
%F A280412 a(n) = 5*a(n-2) - 4*a(n-4).
%F A280412 (End)
%F A280412 Conjectures from _Colin Barker_, Jan 03 2017: (Start)
%F A280412 a(n) = 1 for n even.
%F A280412 a(n) = 2^(n+1) - 2 for n odd.
%F A280412 (End)
%t A280412 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A280412 code = 261; stages = 128;
%t A280412 rule = IntegerDigits[code, 2, 10];
%t A280412 g = 2 * stages + 1; (* Maximum size of grid *)
%t A280412 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A280412 ca = a;
%t A280412 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A280412 PrependTo[ca, a];
%t A280412 (* Trim full grid to reflect growth by one cell at each stage *)
%t A280412 k = (Length[ca[[1]]] + 1)/2;
%t A280412 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 A280412 Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 2], {i, 1, stages - 1}]
%Y A280412 Cf. A280410, A280411, A051049.
%K A280412 nonn,easy
%O A280412 0,2
%A A280412 _Robert Price_, Jan 02 2017