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A277797
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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 1", based on the 5-celled von Neumann neighborhood.
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4
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1, 0, 1, 1100, 1, 111100, 1, 11111100, 1, 1111111100, 1, 111111111100, 1, 11111111111100, 1, 1111111111111100, 1, 111111111111111100, 1, 11111111111111111100, 1, 1111111111111111111100, 1, 111111111111111111111100, 1, 11111111111111111111111100, 1
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OFFSET
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0,4
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COMMENTS
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Initialized with a single black (ON) cell at stage zero.
Rule numbers 1, 9, 17, 25, 257, 265, 273 and 281 all generate this sequence.
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
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LINKS
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Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
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FORMULA
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Conjectures from Colin Barker, Nov 01 2016: (Start)
G.f.: (1 - 100*x^2 + 1100*x^3)/((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
a(n) = 101*a(n-2) - 100*a(n-4) for n>3.
a(n) = (-91+109*(-1)^n+10^(1+n)-(-1)^n*10^(1+n))/18. (End)
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MATHEMATICA
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CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=1; stages=128;
rule=IntegerDigits[code, 2, 10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}}, {g, g}, 0, Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule, ca], {n, 1, stages+1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]], Range[k+1-n, k-1+n]], {j, k+1-n, k-1+n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]][[i]], Range[1, i]], 10], {i, 1, stages-1}]
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CROSSREFS
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Cf. A277798, A277799, A277800.
Sequence in context: A252418 A261911 A276708 * A278466 A280973 A260592
Adjacent sequences: A277794 A277795 A277796 * A277798 A277799 A277800
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KEYWORD
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nonn,easy
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AUTHOR
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Robert Price, Oct 31 2016
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STATUS
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approved
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