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A290192
Binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.
4
1, 10, 101, 1101, 11100, 111101, 1111100, 11111101, 111111100, 1111111101, 11111111100, 111111111101, 1111111111100, 11111111111101, 111111111111100, 1111111111111101, 11111111111111100, 111111111111111101, 1111111111111111100, 11111111111111111101
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.
FORMULA
From Chai Wah Wu, Nov 01 2018: (Start)
a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n > 5 (conjectured).
G.f.: (10*x^5 + 89*x^4 + 91*x^3 + 1)/((x - 1)*(x + 1)*(10*x - 1)) (conjectured). (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 705; 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, Jul 23 2017
STATUS
approved