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A282003
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 413", based on the 5-celled von Neumann neighborhood.
4
1, 1, 110, 111, 11000, 11111, 1100000, 1111111, 110000000, 111111111, 11000000000, 11111111111, 1100000000000, 1111111111111, 110000000000000, 111111111111111, 11000000000000000, 11111111111111111, 1100000000000000000, 1111111111111111111
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.
FORMULA
Conjectures from Colin Barker, Feb 05 2017: (Start)
a(n) = (89*(-10)^n + 109*10^n) / 180 for n>0 and even.
a(n) = (89*(-10)^n + 109*10^n - 20) / 180 for n odd.
a(n) = 101*a(n-2) - 100*a(n-4) for n>4.
G.f.: (1 + x - x^2)*(1 + 10*x^2) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 413; 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[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Feb 04 2017
STATUS
approved