cp's OEIS Frontend

A286519 Binary representation of the diagonal from the origin to the corner (or of the corner to the origin) of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 659", based on the 5-celled von Neumann neighborhood.

%I A286519 #16 Feb 16 2025 08:33:45
%S A286519 1,11,101,1111,11111,111111,1111111,11111111,111111111,1111111111,
%T A286519 11111111111,111111111111,1111111111111,11111111111111,
%U A286519 111111111111111,1111111111111111,11111111111111111,111111111111111111,1111111111111111111,11111111111111111111
%N A286519 Binary representation of the diagonal from the origin to the corner (or of the corner to the origin) of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 659", based on the 5-celled von Neumann neighborhood.
%C A286519 Initialized with a single black (ON) cell at stage zero.
%D A286519 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A286519 Robert Price, <a href="/A286519/b286519.txt">Table of n, a(n) for n = 0..126</a>
%H A286519 Robert Price, <a href="/A286519/a286519.tmp.txt">Diagrams of first 20 stages</a>
%H A286519 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 A286519 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A286519 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A286519 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A286519 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A286519 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A286519 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A286519 Conjectures from _Colin Barker_, Jul 22 2017: (Start)
%F A286519 G.f.: (1 - 10*x^2 + 110*x^3 - 100*x^4) / ((1 - x)*(1 - 10*x)).
%F A286519 a(n) = (10^(1+n) - 1) / 9 for n>2.
%F A286519 a(n) = 11*a(n-1) - 10*a(n-2) for n>4.
%F A286519 (End)
%t A286519 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A286519 code = 659; stages = 128;
%t A286519 rule = IntegerDigits[code, 2, 10];
%t A286519 g = 2 * stages + 1; (* Maximum size of grid *)
%t A286519 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A286519 ca = a;
%t A286519 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A286519 PrependTo[ca, a];
%t A286519 (* Trim full grid to reflect growth by one cell at each stage *)
%t A286519 k = (Length[ca[[1]]] + 1)/2;
%t A286519 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 A286519 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A286519 Cf. A286518, A286520, A286521.
%K A286519 nonn,easy
%O A286519 0,2
%A A286519 _Robert Price_, Jul 22 2017