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A183060
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Number of "ON" cells at n-th stage in a simple 2-dimensional cellular automaton (see Comments for precise definition).
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4
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0, 1, 4, 7, 14, 17, 24, 31, 50, 53, 60, 67, 86, 93, 112, 131, 186, 189, 196, 203, 222, 229, 248, 267, 322, 329, 348, 367, 422, 441, 496, 551, 714, 717, 724, 731, 750, 757, 776, 795, 850, 857, 876, 895, 950, 969, 1024, 1079, 1242, 1249, 1268, 1287
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OFFSET
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0,3
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COMMENTS
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On the semi-infinite square grid, start with all cells OFF.
Turn a single cell to the ON state in row 1.
At each subsequent step, each cell with exactly one neighbor ON is turned ON, and everything that is already ON remains ON.
The sequence gives the number of "ON" cells after n stages. A183061 (the first differences) gives the number of cells turned "ON" at the n-th stage.
Note that this is just half plus the rest of the center line of the cellular automaton described in A147652.
After 2^k stages the structure resembles an isosceles right triangle. For a three-dimensional version using cubes see A186410. For more information see A147562.
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LINKS
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FORMULA
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a(n) = n + (A147562(n) - 1)/2, n >= 1.
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EXAMPLE
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Illustration of the structure after eight stages in which we label the generations of cells turned ON by consecutive numbers:
8
878
8 6 8
8765678
8 8 4 8 8
878 434 878
8 6 4 2 4 6 8
876543212345678
...................
There are 50 "ON" cells so a(8) = 50.
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MATHEMATICA
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A183060[0] = 0; A183060[n_] := Total[With[{m = n - 1}, CellularAutomaton[{4042387958, 2, {{0, 1}, {-1, 0}, {0, 0}, {1, 0}, {0, -1}}}, {{{1}}, 0}, {{{m}}, -m}]], 2] (* JungHwan Min, Jan 24 2016 *)
A183060[0] = 0; A183060[n_] := Total[With[{m = n - 1}, CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, {{{m}}, -m}]], 2] (* JungHwan Min, Jan 24 2016 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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