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A278901
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Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 107", based on the 5-celled von Neumann neighborhood.
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4
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1, 2, 4, 15, 4, 59, 0, 239, 80, 943, 320, 3839, 1344, 15039, 5376, 60415, 21760, 240383, 87040, 962559, 349184, 3845119, 1396736, 15384575, 5591040, 61517823, 22364160, 246087679, 89473024, 984268799, 357892096, 3937140735, 1431633920, 15748235263
(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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a(n) = 5*a(n-2) - 20*a(n-6) + 16*a(n-8) for n>15.
G.f.: (1 +2*x -x^2 +5*x^3 -16*x^4 -16*x^5 -16*x^7 +144*x^8 +16*x^9 -64*x^10 +64*x^11 -320*x^12 -320*x^13 +256*x^14 +256*x^15) / ((1 -x)*(1 +x)*(1 -2*x)*(1 +2*x)*(1 -2*x^2)*(1 +2*x^2)).
(End)
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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=107; 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]], 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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