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A290662 Decimal representation of the diagonal from the origin to the corner 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 A290662 #14 Feb 16 2025 08:33:50
%S A290662 1,3,5,15,23,63,95,255,383,1023,1535,4095,6143,16383,24575,65535,
%T A290662 98303,262143,393215,1048575,1572863,4194303,6291455,16777215,
%U A290662 25165823,67108863,100663295,268435455,402653183,1073741823,1610612735,4294967295,6442450943
%N A290662 Decimal representation of the diagonal from the origin to the corner 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 A290662 Initialized with a single black (ON) cell at stage zero.
%D A290662 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A290662 Robert Price, <a href="/A290662/b290662.txt">Table of n, a(n) for n = 0..126</a>
%H A290662 Robert Price, <a href="/A290662/a290662.tmp.txt">Diagrams of first 20 stages</a>
%H A290662 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 A290662 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A290662 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A290662 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A290662 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A290662 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A290662 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A290662 Conjectures from _Chai Wah Wu_, Aug 03 2020: (Start)
%F A290662 a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n > 3.
%F A290662 G.f.: (2*x^3 - 2*x^2 + 2*x + 1)/((x - 1)*(2*x - 1)*(2*x + 1)). (End)
%t A290662 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A290662 code = 899; stages = 128;
%t A290662 rule = IntegerDigits[code, 2, 10];
%t A290662 g = 2 * stages + 1; (* Maximum size of grid *)
%t A290662 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A290662 ca = a;
%t A290662 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A290662 PrependTo[ca, a];
%t A290662 (* Trim full grid to reflect growth by one cell at each stage *)
%t A290662 k = (Length[ca[[1]]] + 1)/2;
%t A290662 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 A290662 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A290662 Cf. A166956, A290660, A290661.
%K A290662 nonn,easy
%O A290662 0,2
%A A290662 _Robert Price_, Aug 08 2017