cp's OEIS Frontend

A283641 Binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 678", based on the 5-celled von Neumann neighborhood.

%I A283641 #10 Feb 16 2025 08:33:42
%S A283641 1,11,101,1101,10101,110101,1010101,11010101,101010101,1101010101,
%T A283641 10101010101,110101010101,1010101010101,11010101010101,
%U A283641 101010101010101,1101010101010101,10101010101010101,110101010101010101,1010101010101010101,11010101010101010101
%N A283641 Binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 678", based on the 5-celled von Neumann neighborhood.
%C A283641 Initialized with a single black (ON) cell at stage zero.
%D A283641 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A283641 Robert Price, <a href="/A283641/b283641.txt">Table of n, a(n) for n = 0..126</a>
%H A283641 Robert Price, <a href="/A283641/a283641.tmp.txt">Diagrams of first 20 stages</a>
%H A283641 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 A283641 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A283641 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A283641 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A283641 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A283641 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A283641 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A283641 Conjectures from _Colin Barker_, Mar 14 2017: (Start)
%F A283641 G.f.: (1 + 10*x - 10*x^2) / ((1 - x)*(1 - 10*x)*(1 + 10*x)).
%F A283641 a(n) = (-2 - 9*(-10)^n + 209*10^n) / 198.
%F A283641 a(n) = a(n-1) + 100*a(n-2) - 100*a(n-3) for n>2.
%F A283641 (End)
%t A283641 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A283641 code = 678; stages = 128;
%t A283641 rule = IntegerDigits[code, 2, 10];
%t A283641 g = 2 * stages + 1; (* Maximum size of grid *)
%t A283641 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A283641 ca = a;
%t A283641 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A283641 PrependTo[ca, a];
%t A283641 (* Trim full grid to reflect growth by one cell at each stage *)
%t A283641 k = (Length[ca[[1]]] + 1)/2;
%t A283641 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 A283641 Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 10], {i, 1, stages - 1}]
%Y A283641 Cf. A266508, A086893, A283642.
%K A283641 nonn,easy
%O A283641 0,2
%A A283641 _Robert Price_, Mar 12 2017