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A271904
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Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 427", based on the 5-celled von Neumann neighborhood.
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1
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1, 6, 19, 51, 88, 177, 257, 402, 515, 807, 976, 1333, 1577, 2158, 2451, 3139, 3536, 4505, 4977, 6082, 6667, 8143, 8816, 10437, 11265, 13334, 14243, 16499, 17592, 20353, 21553, 24530, 25923, 29495, 31008, 34805, 36553, 41022, 42883, 47619, 49744, 55209, 57473
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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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Empirical g.f.: (1 +6*x +18*x^2 +44*x^3 +61*x^4 +96*x^5 +88*x^6 +68*x^7 +7*x^8 +26*x^9 +6*x^10 -5*x^12) / ((1 -x)^4*(1 +x)^3*(1 +x^2)^2*(1 +x +x^2)). - Colin Barker, Apr 21 2016
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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=427; 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}];
on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
Table[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum 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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