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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

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

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[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 *)

PROG

(PARI) A160410(n)=sum(i=0, n-1, 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: A121054 A273368 A209979 * 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

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Last modified May 25 12:55 EDT 2016. Contains 273290 sequences.