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A256266 Total number of ON states after n generations of cellular automaton based on triangles (see Comments lines for definition). 3

%I

%S 0,6,18,24,48,66,78,84,132,174,210,240,264,282,294,300,396,486,570,

%T 648,720,786,846,900,948,990,1026,1056,1080,1098,1110,1116,1308,1494,

%U 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

%N Total number of ON states after n generations of cellular automaton based on triangles (see Comments lines for definition).

%C On the infinite triangular grid we start at stage 0 with a hexagon formed by six OFF cells, so a(0) = 0.

%C 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.

%C We use the same rules as A255748 for every one of the six 60-degree wedges of the structure.

%C If n is a power of 2 minus 1 and n is greater than 2, then the structure looks like concentric six-pointed stars.

%C 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.

%C Note that in every wedge the structure seems to grow into the holes of a virtual SierpiƄski's triangle (see example).

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%F a(n) = 6 * A255748(n), n >= 1.

%e Illustration of the structure after 15 generations:

%e (Note that every circle should be replaced with a triangle.)

%e .

%e . O

%e . O O

%e . O O O

%e . O O O O

%e . O O O O O

%e . O O O O O O

%e . O O O O O O O

%e . O O O O O O O O

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%e . O O O O O O O \ O O / O O O O O O O

%e . O O O O O O \ O O O / O O O O O O

%e . O O O O O \ O O O O / O O O O O

%e . O O O O O O O O \ O / O O O O O O O O

%e . O O O O O O \ O O / O O O O O O

%e . O O O O O O \ O / O O O O O O

%e . O O O O \ / O O O O

%e . - - - - - - - - - - - - - - - -

%e . O O O O / \ O O O O

%e . O O O O O O / O \ O O O O O O

%e . O O O O O O / O O \ O O O O O O

%e . O O O O O O O O / O \ O O O O O O O O

%e . O O O O O / O O O O \ O O O O O

%e . O O O O O O / O O O \ O O O O O O

%e . O O O O O O O / O O \ O O O O O O O

%e . O O O O O O O O / O \ O O O O O O O O

%e . O O O O O O O O

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%e . O O

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%e .

%e There are 300 ON cells, so a(15) = 300.

%Y Cf. A001316, A047999, A080079, A139250, A151723, A160120, A161330, A161644, A255748, A256256.

%K nonn,look

%O 0,2

%A _Omar E. Pol_, Mar 20 2015

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Last modified October 16 20:35 EDT 2019. Contains 328103 sequences. (Running on oeis4.)