cp's OEIS Frontend

A289404 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 566", based on the 5-celled von Neumann neighborhood.

%I A289404 #31 Feb 16 2025 08:33:48
%S A289404 1,1,11,11,111,111,1111,1111,11111,11111,111111,111111,1111111,
%T A289404 1111111,11111111,11111111,111111111,111111111,1111111111,1111111111,
%U A289404 11111111111,11111111111,111111111111,111111111111,1111111111111,1111111111111,11111111111111
%N A289404 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 566", based on the 5-celled von Neumann neighborhood.
%C A289404 Initialized with a single black (ON) cell at stage zero.
%D A289404 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A289404 Robert Price, <a href="/A289404/b289404.txt">Table of n, a(n) for n = 0..126</a>
%H A289404 Robert Price, <a href="/A289404/a289404.tmp.txt">Diagrams of first 20 stages</a>
%H A289404 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 A289404 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A289404 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A289404 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A289404 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A289404 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A289404 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A289404 Conjectures from _Colin Barker_, Jul 05 2017: (Start)
%F A289404 G.f.: 1 / ((1 - x)*(1 - 10*x^2)).
%F A289404 a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n>2.
%F A289404 (End)
%F A289404 Conjectures from _Federico Provvedi_, Nov 21 2018: (Start)
%F A289404 a(n) = (10^(1 + floor(n/2)) - 1)/9.
%F A289404 a(n) = (sqrt(10)^(n+1)*((sqrt(10)-1)*(-1)^n+(sqrt(10)+1))-2)/18.
%F A289404 (End)
%t A289404 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A289404 code = 566; stages = 128;
%t A289404 rule = IntegerDigits[code, 2, 10];
%t A289404 g = 2 * stages + 1; (* Maximum size of grid *)
%t A289404 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A289404 ca = a;
%t A289404 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A289404 PrependTo[ca, a];
%t A289404 (* Trim full grid to reflect growth by one cell at each stage *)
%t A289404 k = (Length[ca[[1]]] + 1)/2;
%t A289404 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 A289404 Table[FromDigits[Table[ca[[i, j, j]], {j, 1, i}], 10], {i, 1, stages - 1}]
%Y A289404 Cf. A289405, A052551, A032085.
%K A289404 nonn,easy
%O A289404 0,3
%A A289404 _Robert Price_, Jul 05 2017