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A280373
Decimal 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 260", based on the 5-celled von Neumann neighborhood.
4
1, 2, 5, 8, 16, 40, 84, 130, 257, 640, 1344, 2080, 4096, 10240, 21504, 33280, 65536, 163840, 344064, 532480, 1048576, 2621440, 5505024, 8519680, 16777216, 41943040, 88080384, 136314880, 268435456, 671088640, 1409286144, 2181038080, 4294967296, 10737418240
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
Conjectures from Colin Barker, Jan 02 2017: (Start)
a(n) = 16*a(n-4) for n>3.
G.f.: (1 + 2*x + 5*x^2 + 8*x^3 + 8*x^5 + 4*x^6 + 2*x^7 + x^8 - 16*x^12) / ((1 - 2*x)*(1 + 2*x)*(1 + 4*x^2)).
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 260; 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]], 2], {i, 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Jan 01 2017
STATUS
approved