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A284353 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 899", based on the 5-celled von Neumann neighborhood.

%I A284353 #31 Feb 16 2025 08:33:43
%S A284353 1,1,7,13,31,61,127,253,511,1021,2047,4093,8191,16381,32767,65533,
%T A284353 131071,262141,524287,1048573,2097151,4194301,8388607,16777213,
%U A284353 33554431,67108861,134217727,268435453,536870911,1073741821,2147483647,4294967293,8589934591
%N A284353 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 899", based on the 5-celled von Neumann neighborhood.
%C A284353 Initialized with a single black (ON) cell at stage zero.
%C A284353 If one begins the Generalized Jacobsthal numbers (A083579) with a(0)=1, instead of a(0)=0, the same sequence will be obtained. - _Henrik Lipskoch_, Jan 28 2021
%D A284353 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A284353 Robert Price, <a href="/A284353/b284353.txt">Table of n, a(n) for n = 0..126</a>
%H A284353 Robert Price, <a href="/A284353/a284353.tmp.txt">Diagrams of first 20 stages</a>
%H A284353 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 A284353 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A284353 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A284353 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A284353 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A284353 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A284353 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A284353 Conjectures from _Colin Barker_, Mar 26 2017: (Start)
%F A284353 G.f.: (1 - x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
%F A284353 a(n) = 2^(n+1) - 1 for n even.
%F A284353 a(n) = 2^(n+1) - 3 for n odd.
%F A284353 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. (End)
%F A284353 Conjecture: For n > 0, a(n) = Sum_{k=0..n-1} C(n,k) * (2-(-1)^k). - _Wesley Ivan Hurt_, Sep 23 2017
%F A284353 Apparently, a(n) = 6*A000975(n-1) + 1 for n >= 1. - _Hugo Pfoertner_, Jan 28 2021
%t A284353 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A284353 code = 899; stages = 128;
%t A284353 rule = IntegerDigits[code, 2, 10];
%t A284353 g = 2 * stages + 1; (* Maximum size of grid *)
%t A284353 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A284353 ca = a;
%t A284353 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A284353 PrependTo[ca, a];
%t A284353 (* Trim full grid to reflect growth by one cell at each stage *)
%t A284353 k = (Length[ca[[1]]] + 1)/2;
%t A284353 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 A284353 Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 2], {i, 1, stages - 1}]
%Y A284353 Cf. A000975, A284351, A284352, A284354.
%K A284353 nonn,easy
%O A284353 0,3
%A A284353 _Robert Price_, Mar 25 2017