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%I A284354 #10 Mar 26 2017 08:58:57
%S A284354 1,2,7,11,31,47,127,191,511,767,2047,3071,8191,12287,32767,49151,
%T A284354 131071,196607,524287,786431,2097151,3145727,8388607,12582911,
%U A284354 33554431,50331647,134217727,201326591,536870911,805306367,2147483647,3221225471,8589934591
%N A284354 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 899", based on the 5-celled von Neumann neighborhood.
%C A284354 Initialized with a single black (ON) cell at stage zero.
%D A284354 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A284354 Robert Price, <a href="/A284354/b284354.txt">Table of n, a(n) for n = 0..126</a>
%H A284354 Robert Price, <a href="/A284354/a284354.tmp.txt">Diagrams of first 20 stages</a>
%H A284354 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 A284354 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A284354 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A284354 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A284354 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A284354 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A284354 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A284354 Conjectures from _Colin Barker_, Mar 26 2017: (Start)
%F A284354 G.f.: (1 + x + x^2) / ((1 - x)*(1 - 2*x)*(1 + 2*x)).
%F A284354 a(n) = (2^(n+3) - 4)/4 for n even.
%F A284354 a(n) = (3*2^(n+1) - 4)/4 for n odd.
%F A284354 a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n>2.
%F A284354 (End)
%t A284354 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A284354 code = 899; stages = 128;
%t A284354 rule = IntegerDigits[code, 2, 10];
%t A284354 g = 2 * stages + 1; (* Maximum size of grid *)
%t A284354 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A284354 ca = a;
%t A284354 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A284354 PrependTo[ca, a];
%t A284354 (* Trim full grid to reflect growth by one cell at each stage *)
%t A284354 k = (Length[ca[[1]]] + 1)/2;
%t A284354 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 A284354 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i ,1, stages - 1}]
%Y A284354 Cf. A284351, A284352, A284353.
%K A284354 nonn,easy
%O A284354 0,2
%A A284354 _Robert Price_, Mar 25 2017

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