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A273309
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Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 641", based on the 5-celled von Neumann neighborhood.
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4
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1, 4, 17, 40, 73, 112, 161, 216, 281, 352, 433, 520, 617, 720, 833, 952, 1081, 1216, 1361, 1512, 1673, 1840, 2017, 2200, 2393, 2592, 2801, 3016, 3241, 3472, 3713, 3960, 4217, 4480, 4753, 5032, 5321, 5616, 5921, 6232, 6553, 6880, 7217, 7560, 7913, 8272, 8641
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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) = (-15+(-1)^n+8*n+8*n^2)/2 for n>1.
a(n) = 4*n^2+4*n-7 for n>1 and even.
a(n) = 4*n^2+4*n-8 for n>1 and odd.
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>3.
G.f.: (1+2*x+9*x^2+8*x^3-4*x^5) / ((1-x)^3*(1+x)).
(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=641; 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}];
Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
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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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