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A285774 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 84", based on the 5-celled von Neumann neighborhood.

%I A285774 #8 Feb 16 2025 08:33:44
%S A285774 1,0,2,0,4,0,40,0,16,0,160,0,1088,0,10880,0,256,0,2560,0,17408,0,
%T A285774 174080,0,1052672,0,10526720,0,71581696,0,715816960,0,65536,0,655360,
%U A285774 0,4456448,0,44564480,0,269484032,0,2694840320,0,18324914176,0,183249141760,0
%N A285774 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 84", based on the 5-celled von Neumann neighborhood.
%C A285774 Initialized with a single black (ON) cell at stage zero.
%D A285774 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A285774 Robert Price, <a href="/A285774/b285774.txt">Table of n, a(n) for n = 0..126</a>
%H A285774 Robert Price, <a href="/A285774/a285774.tmp.txt">Diagrams of first 20 stages</a>
%H A285774 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 A285774 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A285774 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A285774 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A285774 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A285774 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A285774 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%t A285774 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A285774 code = 84; stages = 128;
%t A285774 rule = IntegerDigits[code, 2, 10];
%t A285774 g = 2 * stages + 1; (* Maximum size of grid *)
%t A285774 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A285774 ca = a;
%t A285774 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A285774 PrependTo[ca, a];
%t A285774 (* Trim full grid to reflect growth by one cell at each stage *)
%t A285774 k = (Length[ca[[1]]] + 1)/2;
%t A285774 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 A285774 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A285774 Cf. A285771, A285772, A285773.
%K A285774 nonn,easy
%O A285774 0,3
%A A285774 _Robert Price_, Apr 25 2017