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A160410
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Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).
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22
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0, 4, 16, 28, 64, 76, 112, 148, 256, 268, 304, 340, 448, 484, 592, 700, 1024, 1036, 1072, 1108, 1216, 1252, 1360, 1468, 1792, 1828, 1936, 2044, 2368, 2476, 2800, 3124, 4096, 4108, 4144, 4180, 4288, 4324, 4432, 4540, 4864, 4900, 5008, 5116, 5440, 5548, 5872, 6196
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OFFSET
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0,2
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COMMENTS
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On the infinite square grid, we consider cells to be the squares, and we start at round 0 with all cells in the OFF state, so a(0) = 0.
At round 1, we turn ON four cells, forming a square.
The rule for n > 1: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells.
Therefore:
At Round 2, we turn ON twelve cells around the square.
At round 3, we turn ON twelve other cells. Three cells around of every corner of the square.
And so on.
For the first differences see the entry A161411.
Shows a fractal behavior similar to the toothpick sequence A139250.
A very similar sequence is A160414, which uses the same rule but with a(1) = 1, not 4.
When n=2^k then the polygon formed by ON cells is a square with side length 2^(k+1).
a(n) is also the area of the figure of A147562 after n generations if A147562 is drawn as overlapping squares. - Omar E. Pol, Nov 08 2009
Also, toothpick sequence starting with four toothpicks centered at (0,0) as a cross.
Rule: Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of three toothpicks of new generation. (Note that these three toothpicks looks like a T-toothpick, see A160172.)
The sequence gives the number of toothpicks after n stages. A161411 gives the number of toothpicks added at the n-th stage.
(End)
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LINKS
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FORMULA
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a(2^k) = (2*(2^k))^2 for k>=0.
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EXAMPLE
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With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
4;
16;
28, 64;
76, 112, 148, 256;
268, 304, 340, 448, 484, 592, 700, 1024;
...
Right border gives the elements of A000302 greater than 1.
This triangle T(n,k) shares with the triangle A256534 the terms of the column k, if k is a power of 2, for example, both triangles share the following terms: 4, 16, 28, 64, 76, 112, 256, 268, 304, 448, 1024, etc.
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Illustration of initial terms, for n = 1..10:
. _ _ _ _ _ _ _ _
. | _ _ | | _ _ |
. | | _|_|_ _ _ _ _ _ _ _ _ _ _ _|_|_ | |
. | |_| _ _ _ _ _ _ _ _ |_| |
. |_ _| | _|_ _|_ | | _|_ _|_ | |_ _|
. | |_| _ _ |_| |_| _ _ |_| |
. | | | _|_|_ _ _ _|_|_ | | |
. | _| |_| _ _ _ _ |_| |_ |
. | | |_ _| | _|_ _|_ | |_ _| | |
. | |_ _| | |_| _ _ |_| | |_ _| |
. | | | | | | | |
. | _ _ | _| |_ _| |_ | _ _ |
. | | _|_| | |_ _ _ _| | |_|_ | |
. | |_| _| |_ _| |_ _| |_ |_| |
. | | | |_ _ _ _ _ _ _ _| | | |
. | _| |_ _| |_ _| |_ _| |_ |
. _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
. | _| |_ _| |_ _| |_ _| |_ _| |_ |
. | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
. | |_ _| | | |_ _| |
. |_ _ _ _| |_ _ _ _|
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After 10 generations there are 304 ON cells, so a(10) = 304.
(End)
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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];
rule=1340761804646523638425234105559798690663900360577570370705802859623\
705267234688669629039040624964794287326910250673678735142700520276191850\
5902735959769690
Show[GraphicsArray[Map[RasterGraphics, CellularAutomaton[{rule, {2,
{{4, 2, 1}, {32, 16, 8}, {256, 128, 64}}}, {1, 1}}, {{{1, 1}, {1, 1}}, 0}, 9, -10]]]];
ca=CellularAutomaton[{rule, {2, {{4, 2, 1}, {32, 16, 8}, {256, 128, 64}}}, {1,
1}}, {{{1, 1}, {1, 1}}, 0}, 99, -100];
Table[Total[ca[[i]], 2], {i, 1, Length[ca]}]
a[n_] := 4*Sum[3^DigitCount[k, 2, 1], {k, 0, n-1}];
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PROG
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CROSSREFS
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Cf. A000079, A000302, A048883, A139250, A147562, A160118, A160412, A160414, A161411, A160717, A160720, A160727, A256530, A256534.
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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