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A151725
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Number of ON states after n generations of cellular automaton based on square grid with each cell adjacent to its eight neighbors.
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15
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0, 1, 9, 13, 33, 37, 57, 77, 121, 125, 145, 165, 209, 237, 297, 373, 465, 469, 489, 509, 553, 581, 641, 717, 809, 837, 897, 981, 1097, 1213, 1409, 1645, 1833, 1837, 1857, 1877, 1921, 1949, 2009, 2085, 2177, 2205, 2265, 2349, 2465, 2581, 2777, 3013
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OFFSET
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0,3
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COMMENTS
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A cell is turned ON if exactly one of its eight neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
Analog of A147562, which is the case when each cell has only four neighbors.
The equivalent Mathematica cellular automaton is obtained with neighborhood weights {{1,1,1},{1,9,1},{1,1,1}}, rule number 261634, and starting configuration {{1}}. [John W. Layman, Sep 11 2009]
Observation: Visual pattern similar to the toothpick structure (see A139250). [Omar E. Pol, Dec 14 2009]
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LINKS
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David Applegate, Table of n, a(n) for n = 0..1000
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. H. Packard and S. Wolfram, Two-Dimensional Cellular Automata, Journal of Statistical Physics, 38 (1985), 901-946. (See page 920, Figure 7e). Alternative copy
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
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FORMULA
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For a recurrence see the Applegate-Pol-Sloane paper.
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MATHEMATICA
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RasterGraphics[state_?MatrixQ, colors_Integer : 2, opts___] := Graphics[Raster[ Reverse[1 - state/(colors - 1)]], AspectRatio -> (AspectRatio /. {opts} /. AspectRatio -> Automatic), Frame -> True, FrameTicks -> None, GridLines -> None]; wt = {{1, 1, 1}, {1, 9, 1}, {1, 1, 1}}; rule= 261634; init={{1}}; Show[GraphicsArray[Map[RasterGraphics, CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, 9, -10]]]]; nx = 100; ca = CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, nx - 1, -nx]; a = Table[Total[ca[[i]], 2], {i, 1, nx}] (* John W. Layman, Sep 11 2009 *)
A151725[0] = 0; A151725[n_] := Total[CellularAutomaton[{174766, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{{n - 1}}}], 2]; Array[A151725, 48, 0] (* JungHwan Min, Sep 01 2016 *)
A151725L[n_] := Prepend[Total[#, 2] & /@ CellularAutomaton[{174766, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, n - 1], 0]; A151725L[47] (* JungHwan Min, Sep 01 2016 *)
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CROSSREFS
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Cf. A139250, A151726, A147562, A147582, A151723, A151735, A151747, A151728.
See A151731, A151732, A151733, A151734 for the same CA except that two neighbors must be ON for a cell to turn ON.
Sequence in context: A098074 A032361 A031196 * A146864 A287766 A146739
Adjacent sequences: A151722 A151723 A151724 * A151726 A151727 A151728
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KEYWORD
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nonn
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AUTHOR
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David Applegate and N. J. A. Sloane, Jun 13 2009
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STATUS
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approved
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