cp's OEIS Frontend

A290192 Binary 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 705", based on the 5-celled von Neumann neighborhood.

%I A290192 #12 Feb 16 2025 08:33:49
%S A290192 1,10,101,1101,11100,111101,1111100,11111101,111111100,1111111101,
%T A290192 11111111100,111111111101,1111111111100,11111111111101,
%U A290192 111111111111100,1111111111111101,11111111111111100,111111111111111101,1111111111111111100,11111111111111111101
%N A290192 Binary 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 705", based on the 5-celled von Neumann neighborhood.
%C A290192 Initialized with a single black (ON) cell at stage zero.
%D A290192 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A290192 Robert Price, <a href="/A290192/b290192.txt">Table of n, a(n) for n = 0..126</a>
%H A290192 Robert Price, <a href="/A290192/a290192.tmp.txt">Diagrams of first 20 stages</a>
%H A290192 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 A290192 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A290192 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A290192 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A290192 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A290192 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A290192 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A290192 From _Chai Wah Wu_, Nov 01 2018: (Start)
%F A290192 a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n > 5 (conjectured).
%F A290192 G.f.: (10*x^5 + 89*x^4 + 91*x^3 + 1)/((x - 1)*(x + 1)*(10*x - 1)) (conjectured). (End)
%t A290192 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A290192 code = 705; stages = 128;
%t A290192 rule = IntegerDigits[code, 2, 10];
%t A290192 g = 2 * stages + 1; (* Maximum size of grid *)
%t A290192 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A290192 ca = a;
%t A290192 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A290192 PrependTo[ca, a];
%t A290192 (* Trim full grid to reflect growth by one cell at each stage *)
%t A290192 k = (Length[ca[[1]]] + 1)/2;
%t A290192 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 A290192 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A290192 Cf. A290193, A290194, A290195.
%K A290192 nonn,easy
%O A290192 0,2
%A A290192 _Robert Price_, Jul 23 2017