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A282871 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 617", based on the 5-celled von Neumann neighborhood. 4
1, 0, 11, 1, 1110, 110, 111110, 11110, 11111110, 1111110, 1111111110, 111111110, 111111111110, 11111111110, 11111111111110, 1111111111110, 1111111111111110, 111111111111110, 111111111111111110, 11111111111111110, 11111111111111111110, 1111111111111111110 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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, Mar 06 2017: (Start)
G.f.: (1 - x - 89*x^2 + 90*x^3 + 9*x^4 + 100*x^6) / ((1 - x)*(1 - 10*x)*(1 + 10*x)).
a(n) = (-2000 + 99*(-10)^n + 101*10^n) / 1800 for n>3.
a(n) = a(n-1) + 100*a(n-2) - 100*a(n-3) for n>4.
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 617; 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: A283283 A283372 A283400 * A286702 A286078 A286766
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 05 2017
STATUS
approved

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Last modified July 12 10:18 EDT 2024. Contains 374244 sequences. (Running on oeis4.)