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A289768 Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 598", based on the 5-celled von Neumann neighborhood.

%I A289768 #16 May 20 2025 12:53:31
%S A289768 1,1,3,3,5,5,13,13,17,17,59,59,81,81,219,219,257,257,899,899,1349,
%T A289768 1349,3437,3437,4353,4353,15235,15235,20805,20805,56173,56173,65537,
%U A289768 65537,229379,229379,344069,344069,876557,876557,1118225,1118225,3913787,3913787
%N A289768 Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 598", based on the 5-celled von Neumann neighborhood.
%C A289768 Initialized with a single black (ON) cell at stage zero.
%D A289768 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A289768 Robert Price, <a href="/A289768/b289768.txt">Table of n, a(n) for n = 0..126</a>
%H A289768 Robert Price, <a href="/A289768/a289768.tmp.txt">Diagrams of first 20 stages</a>
%H A289768 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 A289768 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A289768 Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A289768 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A289768 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A289768 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A289768 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A289768 a(n) = Sum_{k=0..n} 2^k*(A167630(floor(n/2), k) mod 2). - _Mélika Tebni_, May 20 2025
%t A289768 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A289768 code = 598; stages = 128;
%t A289768 rule = IntegerDigits[code, 2, 10];
%t A289768 g = 2 * stages + 1; (* Maximum size of grid *)
%t A289768 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A289768 ca = a;
%t A289768 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A289768 PrependTo[ca, a];
%t A289768 (* Trim full grid to reflect growth by one cell at each stage *)
%t A289768 k = (Length[ca[[1]]] + 1)/2;
%t A289768 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 A289768 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A289768 Cf. A167630, A289766, A289767, A289769.
%K A289768 nonn,easy
%O A289768 0,3
%A A289768 _Robert Price_, Jul 12 2017