

A160410


Number of "ON" cells at nth stage in simple 2dimensional 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
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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 Ttoothpick, see A160172.)
The sequence gives the number of toothpicks after n stages. A161411 gives the number of toothpicks added at the nth stage.
(End)


LINKS

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), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.],
Omar E. Pol, Illustration of initial terms (2009)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata


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[1state/(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, n1}];
Table[a[n], {n, 0, 50}] (* JeanFrançois Alcover, Nov 17 2017, after N. J. A. Sloane *)


PROG

(PARI) A160410(n)=sum(i=0, n1, 3^norml2(binary(i)))<<2 \\ M. F. Hasler, Dec 04 2012


CROSSREFS

Cf. A000079, A000302, A048883, A139250, A147562, A160118, A160412, A160414, A161411, A160717, A160720, A160727, A256530, A256534.
Sequence in context: A273368 A209979 A294629 * A256534 A227434 A173019
Adjacent sequences: A160407 A160408 A160409 * A160411 A160412 A160413


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



