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A269511
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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 342", based on the 5-celled von Neumann neighborhood.
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1
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1, 6, 14, 38, 55, 108, 140, 236, 285, 434, 506, 722, 819, 1112, 1240, 1624, 1785, 2270, 2470, 3070, 3311, 4036, 4324, 5188, 5525, 6538, 6930, 8106, 8555, 9904, 10416, 11952, 12529, 14262, 14910, 16854, 17575, 19740, 20540, 22940, 23821, 26466, 27434, 30338
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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) = a(n-1)+2*a(n-2)-2*a(n-3)-2*a(n-6)+2*a(n-7)+a(n-8)-a(n-9) for n>8.
G.f.: (1+5*x+6*x^2+14*x^3+x^4+5*x^5) / ((1-x)^4*(1+x)^3*(1+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=342; 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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