cp's OEIS Frontend

A290111 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 643", based on the 5-celled von Neumann neighborhood.

%I A290111 #15 Feb 16 2025 08:33:49
%S A290111 1,11,101,1101,11101,111101,1111101,11111101,111111101,1111111101,
%T A290111 11111111101,111111111101,1111111111101,11111111111101,
%U A290111 111111111111101,1111111111111101,11111111111111101,111111111111111101,1111111111111111101,11111111111111111101
%N A290111 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 643", based on the 5-celled von Neumann neighborhood.
%C A290111 Initialized with a single black (ON) cell at stage zero.
%D A290111 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A290111 Robert Price, <a href="/A290111/b290111.txt">Table of n, a(n) for n = 0..126</a>
%H A290111 Robert Price, <a href="/A290111/a290111.tmp.txt">Diagrams of first 20 stages</a>
%H A290111 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 A290111 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A290111 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A290111 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A290111 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A290111 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A290111 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A290111 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-10).
%F A290111 From _Chai Wah Wu_, Apr 02 2024: (Start)
%F A290111 a(n) = 11*a(n-1) - 10*a(n-2) for n > 3.
%F A290111 G.f.: (100*x^3 - 10*x^2 + 1)/((x - 1)*(10*x - 1)). (End)
%t A290111 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A290111 code = 643; stages = 128;
%t A290111 rule = IntegerDigits[code, 2, 10];
%t A290111 g = 2 * stages + 1; (* Maximum size of grid *)
%t A290111 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A290111 ca = a;
%t A290111 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A290111 PrependTo[ca, a];
%t A290111 (* Trim full grid to reflect growth by one cell at each stage *)
%t A290111 k = (Length[ca[[1]]] + 1)/2;
%t A290111 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 A290111 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A290111 Cf. A290112, A290113, A290114.
%K A290111 nonn,easy
%O A290111 0,2
%A A290111 _Robert Price_, Jul 19 2017