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A273121
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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 593", based on the 5-celled von Neumann neighborhood.
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1
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1, 5, 22, 47, 107, 172, 309, 445, 669, 890, 1235, 1563, 2039, 2508, 3149, 3813, 4609, 5458, 6459, 7527, 8747, 10032, 11525, 13049, 14785, 16582, 18627, 20723, 23059, 25488, 28149, 30973, 34021, 37214, 40570, 44206, 48131, 52084, 56492, 60888, 65753, 70590
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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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Robert Price, Table of n, a(n) for n = 0..128
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
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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=593; 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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Cf. A273119.
Sequence in context: A184724 A082005 A273024 * A099078 A272836 A273575
Adjacent sequences: A273118 A273119 A273120 * A273122 A273123 A273124
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KEYWORD
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nonn,easy
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AUTHOR
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Robert Price, May 15 2016
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STATUS
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approved
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