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A277798 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 1", based on the 5-celled von Neumann neighborhood.

%I A277798 #12 Feb 16 2025 08:33:37
%S A277798 1,0,100,11,10000,1111,1000000,111111,100000000,11111111,10000000000,
%T A277798 1111111111,1000000000000,111111111111,100000000000000,11111111111111,
%U A277798 10000000000000000,1111111111111111,1000000000000000000,111111111111111111,100000000000000000000
%N A277798 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 1", based on the 5-celled von Neumann neighborhood.
%C A277798 Initialized with a single black (ON) cell at stage zero.
%C A277798 Rule numbers 1, 9, 17, 25, 257, 265, 273 and 281 all generate this sequence.
%D A277798 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A277798 Robert Price, <a href="/A277798/b277798.txt">Table of n, a(n) for n = 0..126</a>
%H A277798 Robert Price, <a href="/A277798/a277798.tmp.txt">Diagrams of first 20 stages</a>
%H A277798 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 A277798 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A277798 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A277798 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A277798 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A277798 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A277798 Conjectures from _Colin Barker_, Nov 01 2016: (Start)
%F A277798 G.f.: (1 - x^2 + 11*x^3)/((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
%F A277798 a(n) = 101*a(n-2) - 100*a(n-4) for n>3.
%F A277798 a(n) = (-10+89*(-10)^n+10*(-1)^n+91*10^n)/180. (End)
%t A277798 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A277798 code=1; stages=128;
%t A277798 rule=IntegerDigits[code,2,10];
%t A277798 g=2*stages+1; (* Maximum size of grid *)
%t A277798 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A277798 ca=a;
%t A277798 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A277798 PrependTo[ca,a];
%t A277798 (* Trim full grid to reflect growth by one cell at each stage *)
%t A277798 k=(Length[ca[[1]]]+1)/2;
%t A277798 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 A277798 Table[FromDigits[Part[ca[[i]][[i]],Range[i,2*i-1]],10], {i,1,stages-1}]
%Y A277798 Cf. A277797, A277799, A277800.
%K A277798 nonn,easy
%O A277798 0,3
%A A277798 _Robert Price_, Oct 31 2016