A164982 -id:A164982 - cp's OEIS frontend

A189007 Number of ON cells after n generations of the 2D cellular automaton described in the comments.

1, 4, 8, 16, 16, 32, 32, 64, 32, 64, 64, 128, 64, 128, 128, 256, 64, 128, 128, 256, 128, 256, 256, 512, 128, 256, 256, 512, 256, 512, 512, 1024, 128, 256, 256, 512, 256, 512, 512, 1024, 256, 512, 512, 1024, 512, 1024, 1024, 2048, 256, 512, 512, 1024, 512, 1024, 1024, 2048, 512, 1024, 1024, 2048, 1024, 2048, 2048, 4096, 256, 512, 512, 1024, 512

Offset: 1

John W. Layman, Apr 15 2011

nonn

The cells are the squares of the standard infinite square grid. All cells are initially OFF and a single cell is turned ON at generation 1. At subsequent generations a cell is ON if and only if exactly one East/West neighbor was ON or exactly one North/South neighbor was ON (or BOTH of those conditions) in the previous generation.
The equivalent Mathematica cellular automaton is obtained with neighborhood weights {{0,1,0},{3,0,3},{0,1,0}}, rule number 186, and initial configuration {{1}}.
Also sequence generated by Rule 84 with neighborhood weights {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}. - Robert Price, Mar 11 2016
Conjecture: a(1) = 1; a(n) = 2^A056791(n-1) for n > 1. - Michael De Vlieger, Nov 02 2022

Michael De Vlieger, Table of n, a(n) for n = 1..512
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 32.
John W. Layman, Graphs of the automaton for generations 1-15

Cf. A164982, A165345, A245191.

ca = CellularAutomaton[{186, {2, {{0, 1, 0}, {3, 0, 3}, {0, 1, 0}}}, {1, 1}}, {{{1}}, 0}, 50-1, -50]; Table[Total[ca[[n]], 2], {n, 1, 50}]

It appears that this sequence is the limit of the following process.  Start with {1,4} and repeatedly perform this set of operations: (1) select the second half H of the sequence; (2) append twice the terms of H, then (3) append four times the terms of H.  This gives {1,4} -> {1,4,8,16} -> {1,4,8,16,16,32,32,64} -> {1,4,8,16,16,32,32,64,32,64,64,128,64,128,128,256} -> ...  This has been verified for the first 150 terms.
Comment from N. J. A. Sloane, Jul 21 2014: (Start)
It is not difficult to show that the preceding conjecture is correct. In fact one can give an explicit formula for the n-th term. At generation n >= 2, the configuration of ON cells consists of a set of concentric diamonds (see the illustration). The sizes of the diamonds are given by the (n-2)nd term of A245191. Let N = A245191(n-2) = Sum_{i>=0} b_i*2^i. Then the ON cells form a set of diamonds with edge-lengths i+2 for each b_i = 1. The i-th diamond contains 4*(i+1) ON cells, and the total number of ON cells is therefore a(n) = 4*Sum_i (i+1)*b_i. The b_i are given explicitly in A245191.
For example, if n=11, N = A245191(9) = 544 = 2^5 + 2^9, so b_5 = b_9 = 1, there are two diamonds, of side lengths 7 and 11, containing a total of 4*(6+10) = 64 = a(11) ON cells. (End)

A165345 Number of ON cells after n generations of the 2D cellular automaton described in the comments.

Original entry on oeis.org

1, 5, 9, 25, 29, 41, 53, 105, 113, 129, 141, 193, 205, 241, 285, 433, 453, 481, 497, 553, 569, 609, 653, 801, 829, 881, 917, 1073, 1109, 1217, 1349, 1793, 1845, 1905, 1933, 2001, 2029, 2081, 2129, 2281, 2313, 2369, 2409, 2569, 2609, 2721, 2853, 3297, 3357

Offset: 1

John W. Layman, Sep 15 2009, Sep 16 2009

nonn

The cells are the squares of the standard square grid. All cells are initially OFF and one cell is turned ON at generation 1. At subsequent generations a cell is ON if and only if (1) it was ON, or (2) exactly one of the four nearest side neighbors was ON, or (3) exactly three of the four nearest corner neighbors were ON, in the previous generation

The equivalent Mathematica automaton is obtained with neighborhood weights {{10,2,10},{2,1,2},{10,2,10}}, rule number 755364134566574, and initial configuration {{1}} (see code).

Graphs of the automaton for generations 1-20. [From _John W. Layman_, Sep 15 2009]

Cf. A048883, A072272, A122108, A160410, A164982.

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 = {{10, 2, 10}, {2, 1, 2}, {10, 2, 10}}; rule=755364134566574; init = {{1}}; Show[GraphicsArray[ Map[RasterGraphics, CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, 9, -10]]]]; ca = CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, 99, -100]; a = Table[Total[ca[[i]], 2], {i, 1, 100}]

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A247002 A164982(2^n).

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A189007 Number of ON cells after n generations of the 2D cellular automaton described in the comments.

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A165345 Number of ON cells after n generations of the 2D cellular automaton described in the comments.

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Mathematica