cp's OEIS Frontend

A287468 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 276", based on the 5-celled von Neumann neighborhood.

%I A287468 #15 Feb 16 2025 08:33:46
%S A287468 1,0,11,0,111,0,1111,0,11111,0,111111,0,1111111,0,11111111,0,
%T A287468 111111111,0,1111111111,0,11111111111,0,111111111111,0,1111111111111,
%U A287468 0,11111111111111,0,111111111111111,0,1111111111111111,0,11111111111111111,0,111111111111111111
%N A287468 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 276", based on the 5-celled von Neumann neighborhood.
%C A287468 Initialized with a single black (ON) cell at stage zero.
%D A287468 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A287468 Robert Price, <a href="/A287468/b287468.txt">Table of n, a(n) for n = 0..126</a>
%H A287468 Robert Price, <a href="/A287468/a287468.tmp.txt">Diagrams of first 20 stages</a>
%H A287468 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 A287468 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A287468 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A287468 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A287468 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A287468 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A287468 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A287468 From _Colin Barker_, May 29 2017: (Start)
%F A287468 Conjectures:
%F A287468 G.f.: 1 / ((1 - x)*(1 + x)*(1 - 10*x^2)).
%F A287468 a(n) = (10^(n/2+1) - 1) / 9 for n even.
%F A287468 a(n) = 0 for n odd.
%F A287468 a(n) = 11*a(n-2) - 10*a(n-4) for n>3.
%F A287468 (End)
%t A287468 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A287468 code = 276; stages = 128;
%t A287468 rule = IntegerDigits[code, 2, 10];
%t A287468 g = 2 * stages + 1; (* Maximum size of grid *)
%t A287468 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A287468 ca = a;
%t A287468 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A287468 PrependTo[ca, a];
%t A287468 (* Trim full grid to reflect growth by one cell at each stage *)
%t A287468 k = (Length[ca[[1]]] + 1)/2;
%t A287468 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 A287468 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A287468 Cf. A287469, A077896, A287470.
%K A287468 nonn,easy
%O A287468 0,3
%A A287468 _Robert Price_, May 25 2017