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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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20
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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. [From Omar E. Pol, Nov 08 2009]
Contribution from Omar E. Pol, Mar 28 2011 (Start):
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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Table of n, a(n) for n=0..47.
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
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
O. E. Pol, Illustration of initial terms [From Omar E. Pol, Nov 08 2009]
Index entries for sequences related to cellular automata
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FORMULA
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Equals 4*A130665. This provides an explicit formula for a(n). - N. J. A. Sloane, Jul 13 2009
a(2^k) = (2*(2^k))^2 for k>=0.
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EXAMPLE
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If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
99..............99
988888888888888889
.8778877887788778.
.8766667887666678.
.8865568888655688.
.8865444444445688.
.8766433443346678.
.8778432222348778.
.8888442112448888.
.8888442112448888.
.8778432222348778.
.8766433443346678.
.8865444444445688.
.8865568888655688.
.8766667887666678.
.8778877887788778.
988888888888888889
99..............99
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MATHEMATICA
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(From John W. Layman, Sep 01, 2009; Sep 02, 2009)
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]}]
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PROG
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(PARI) A160410(n)=sum(i=0, n-1, 3^norml2(binary(i)))<<2 \\ - M. F. Hasler, Dec 04 2012
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CROSSREFS
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Cf. A139250, A139251, A160118.
Cf. A000079, A048883, A147582, A160412, A160414, A161411, A160717, A160720, A160727.
Cf. A147562. [From Omar E. Pol, Nov 08 2009]
Sequence in context: A161335 A121054 A209979 * A173019 A031003 A036345
Adjacent sequences: A160407 A160408 A160409 * A160411 A160412 A160413
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KEYWORD
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nonn
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AUTHOR
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Omar E. Pol, May 20 2009, Jun 13 2009
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EXTENSIONS
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Edited by David Applegate and N. J. A. Sloane, Jul 13 2009
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STATUS
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approved
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