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A282103
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 427", based on the 5-celled von Neumann neighborhood.
4
1, 1, 111, 100, 11101, 10111, 1111100, 1000101, 111010111, 101111100, 11111000101, 10001010111, 1110101111100, 1011111000101, 111110001010111, 100010101111100, 11101011111000101, 10111110001010111, 1111100010101111100, 1000101011111000101
OFFSET
0,3
COMMENTS
Initialized with a single black (ON) cell at stage zero.
The initial terms coincide with those of A281894, although the two sequences are different (first disagreement at index 12). - Bruno Berselli, Feb 07 2017
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
FORMULA
Empirical g.f.: (1 + x + 111*x^2 + 99*x^3 + 1100*x^4 + 1000*x^6 - 1000*x^7) / ((1 - x)*(1 - 10*x)*(1 + 10*x)*(1 + x + x^2)*(1 + 100*x^2)). - Colin Barker, Feb 06 2017
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 427; 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
KEYWORD
nonn,easy
AUTHOR
Robert Price, Feb 06 2017
STATUS
approved