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A282482 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 478", based on the 5-celled von Neumann neighborhood. 4
1, 11, 110, 1111, 11000, 111110, 1100011, 11111110, 110000011, 1111101110, 11000111011, 111111101110, 1100000111011, 11111011101110, 110001110111011, 1111111011101110, 11000001110111011, 111110111011101110, 1100011101110111011, 11111110111011101110 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Initialized with a single black (ON) cell at stage zero.
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
LINKS
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
FORMULA
Conjectures from Colin Barker, Feb 18 2017: (Start)
a(n) = a(n-2) + 10000*a(n-4) - 10000*a(n-6) for n>6.
G.f.: (1 + 11*x + 109*x^2 + 1100*x^3 + 890*x^4 - x^5 - 989*x^6 + 1000*x^10) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)*(1 + 100*x^2)).
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 478; 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}]
CROSSREFS
Sequence in context: A282006 A280099 A282385 * A286205 A286500 A287099
KEYWORD
nonn,easy
AUTHOR
Robert Price, Feb 16 2017
STATUS
approved

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Last modified June 19 03:10 EDT 2024. Contains 373492 sequences. (Running on oeis4.)