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A282606
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 489", based on the 5-celled von Neumann neighborhood.
4
1, 0, 111, 100, 11011, 11110, 1110011, 1011010, 110111001, 111101000, 11100111001, 10110101000, 1101110011001, 1111010011000, 111001110011001, 101101010011000, 11011100110011001, 11110100110011000, 1110011100110011001, 1011010100110011000
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 Chai Wah Wu, May 05 2024: (Start)
a(n) = a(n-2) + 10000*a(n-4) - 10000*a(n-6) for n > 13.
G.f.: (10000*x^13 - 10010*x^9 + 990*x^8 - 100*x^7 - 1000*x^6 + 11010*x^5 + 900*x^4 + 100*x^3 + 110*x^2 + 1)/(10000*x^6 - 10000*x^4 - x^2 + 1). (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 489; 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 19 2017
STATUS
approved