0,2

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.

The rule: 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.

Initial condition: At round 1, we turn ON four cells, forming a square.

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-like behavior similar to the toothpick sequence A139250.

A very similar sequence is A160414, which uses the same rule but starts with one ON cell.

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 (info(AT)polprimos.com), Nov 08 2009]

Index entries for sequences related to cellular automata

O. E. Pol, Illustration of initial terms [From Omar E. Pol (info(AT)polprimos.com), Nov 08 2009]

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.

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

(From John W. Layman (layman(AT)math.vt.edu), 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]}]

Cf. A139250, A139251, A160118.

Cf. A000079, A048883, A147582, A160412, A160414, A161411, A160717, A160720, A160727.

Cf. A147562. [From Omar E. Pol (info(AT)polprimos.com), Nov 08 2009]

Sequence in context: A017569 A161335 A121054 this_sequence A031003 A036345 A046001

Adjacent sequences: A160407 A160408 A160409 this_sequence A160411 A160412 A160413

nonn,new

Omar E. Pol (info(AT)polprimos.com), May 20 2009, Jun 13 2009

Edited by David Applegate and N. J. A. Sloane, Jul 13 2009