

A160412


Number of "ON" cells at nth stage in simple 2dimensional cellular automaton (see Comments for precise definition).


8



0, 3, 12, 21, 48, 57, 84, 111, 192, 201, 228, 255, 336, 363, 444, 525, 768, 777, 804, 831, 912, 939, 1020, 1101, 1344, 1371, 1452, 1533, 1776, 1857, 2100, 2343, 3072, 3081, 3108, 3135, 3216, 3243, 3324, 3405, 3648, 3675, 3756, 3837, 4080, 4161, 4404, 4647
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OFFSET

0,2


COMMENTS

On the infinite square grid, consider the outside corner of an infinite square.
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.
At round 1, we turn ON three cells around the corner of the infinite square, forming a concaveconvex hexagon with three exposed vertices.
At round 2, we turn ON nine cells around the hexagon.
At round 3, we turn ON nine other cells. Three cells around of every corner of the hexagon.
And so on.
Shows a fractallike behavior similar to the toothpick sequence A153006.
For the first differences see the entry A162349.
For more information see A160410, which is the main entry for this sequence.
(End)


LINKS



FORMULA

a(0) = 0, a(n) = A130665(n1)*3, for n>0.
(End)


EXAMPLE

If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
...77..77..77..77
...766667..766667
....6556....6556.
....654444444456.
...76643344334667
...77.43222234.77
......44211244...
00000000001244...
00000000002234.77
00000000004334667
0000000000444456.
0000000000..6556.
0000000000.766667
0000000000.77..77
0000000000.......
0000000000.......
0000000000.......


MATHEMATICA

a[n_] := 3*Sum[3^DigitCount[k, 2, 1], {k, 0, n  1}]; Array[a, 48, 0] (* Michael De Vlieger, Nov 01 2022 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



