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A256266
Total number of ON states after n generations of cellular automaton based on triangles (see Comments lines for definition).
4
0, 6, 18, 24, 48, 66, 78, 84, 132, 174, 210, 240, 264, 282, 294, 300, 396, 486, 570, 648, 720, 786, 846, 900, 948, 990, 1026, 1056, 1080, 1098, 1110, 1116, 1308, 1494, 1674, 1848, 2016, 2178, 2334, 2484, 2628, 2766, 2898, 3024, 3144, 3258, 3366, 3468, 3564, 3654, 3738, 3816, 3888, 3954, 4014, 4068, 4116, 4158, 4194, 4224, 4248
OFFSET
0,2
COMMENTS
On the infinite triangular grid we start at stage 0 with a hexagon formed by six OFF cells, so a(0) = 0.
At stage 1, around the mentioned hexagon, six triangular cells connected by their vertices are turned ON forming a six-pointed star, so a(1) = 6.
We use the same rules as A255748 for every one of the six 60-degree wedges of the structure.
If n is a power of 2 minus 1 and n is greater than 2, then the structure looks like concentric six-pointed stars.
If n is a power of 2 and n is greater than 2, then the structure looks like a hexagon that contains concentric six-pointed stars.
Note that in every wedge the structure seems to grow into the holes of a virtual Sierpiński's triangle (see example).
FORMULA
a(n) = 6 * A255748(n), n >= 1.
EXAMPLE
Illustration of the structure after 15 generations:
(Note that every circle should be replaced with a triangle.)
.
. O
. O O
. O O O
. O O O O
. O O O O O
. O O O O O O
. O O O O O O O
. O O O O O O O O
. O O O O O O O O \ O / O O O O O O O O
. O O O O O O O \ O O / O O O O O O O
. O O O O O O \ O O O / O O O O O O
. O O O O O \ O O O O / O O O O O
. O O O O O O O O \ O / O O O O O O O O
. O O O O O O \ O O / O O O O O O
. O O O O O O \ O / O O O O O O
. O O O O \ / O O O O
. - - - - - - - - - - - - - - - -
. O O O O / \ O O O O
. O O O O O O / O \ O O O O O O
. O O O O O O / O O \ O O O O O O
. O O O O O O O O / O \ O O O O O O O O
. O O O O O / O O O O \ O O O O O
. O O O O O O / O O O \ O O O O O O
. O O O O O O O / O O \ O O O O O O O
. O O O O O O O O / O \ O O O O O O O O
. O O O O O O O O
. O O O O O O O
. O O O O O O
. O O O O O
. O O O O
. O O O
. O O
. O
.
There are 300 ON cells, so a(15) = 300.
MATHEMATICA
6*Join[{0}, Accumulate@ Flatten@ Table[Range[2^n, 1, -1], {n, 0, 5}]] (* Michael De Vlieger, Nov 03 2022 *)
KEYWORD
nonn,look
AUTHOR
Omar E. Pol, Mar 20 2015
STATUS
approved