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A160410 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition). 22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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
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)
LINKS
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.],
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. 31.
Omar E. Pol, Illustration of initial terms (2009)
FORMULA
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.
EXAMPLE
From Omar E. Pol, Sep 24 2015: (Start)
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.
.
Illustration of initial terms, for n = 1..10:
. _ _ _ _ _ _ _ _
. | _ _ | | _ _ |
. | | _|_|_ _ _ _ _ _ _ _ _ _ _ _|_|_ | |
. | |_| _ _ _ _ _ _ _ _ |_| |
. |_ _| | _|_ _|_ | | _|_ _|_ | |_ _|
. | |_| _ _ |_| |_| _ _ |_| |
. | | | _|_|_ _ _ _|_|_ | | |
. | _| |_| _ _ _ _ |_| |_ |
. | | |_ _| | _|_ _|_ | |_ _| | |
. | |_ _| | |_| _ _ |_| | |_ _| |
. | | | | | | | |
. | _ _ | _| |_ _| |_ | _ _ |
. | | _|_| | |_ _ _ _| | |_|_ | |
. | |_| _| |_ _| |_ _| |_ |_| |
. | | | |_ _ _ _ _ _ _ _| | | |
. | _| |_ _| |_ _| |_ _| |_ |
. _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
. | _| |_ _| |_ _| |_ _| |_ _| |_ |
. | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
. | |_ _| | | |_ _| |
. |_ _ _ _| |_ _ _ _|
.
After 10 generations there are 304 ON cells, so a(10) = 304.
(End)
MATHEMATICA
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]}]
(* John W. Layman, Sep 01 2009; Sep 02 2009 *)
a[n_] := 4*Sum[3^DigitCount[k, 2, 1], {k, 0, n-1}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 17 2017, after N. J. A. Sloane *)
PROG
(PARI) A160410(n)=sum(i=0, n-1, 3^norml2(binary(i)))<<2 \\ M. F. Hasler, Dec 04 2012
CROSSREFS
Sequence in context: A273368 A209979 A294629 * A256534 A352205 A364582
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, May 20 2009
EXTENSIONS
Edited by David Applegate and N. J. A. Sloane, Jul 13 2009
STATUS
approved

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Last modified March 28 09:03 EDT 2024. Contains 371239 sequences. (Running on oeis4.)