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A277936
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Decimal representation of the x-axis, from the left edge to the origin, or from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 7", based on the 5-celled von Neumann neighborhood.
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2
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1, 3, 0, 15, 0, 63, 0, 255, 0, 1023, 0, 4095, 0, 16383, 0, 65535, 0, 262143, 0, 1048575, 0, 4194303, 0, 16777215, 0, 67108863, 0, 268435455, 0, 1073741823, 0, 4294967295, 0, 17179869183, 0, 68719476735, 0, 274877906943, 0, 1099511627775, 0, 4398046511103, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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Initialized with a single black (ON) cell at stage zero.
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
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LINKS
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FORMULA
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G.f.: (1+3*x-5*x^2+4*x^4) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)).
a(n) = 5*a(n-2)-4*a(n-4) for n>4.
a(n) = (-1/2-(-2)^n+(-1)^n/2+2^n) for n>0. (End)
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MAPLE
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MATHEMATICA
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CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=7; 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}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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