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%I #65 Nov 04 2022 07:31:52
%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 Sierpiński's triangle (see example).
%H Michael De Vlieger, <a href="/A256266/b256266.txt">Table of n, a(n) for n = 0..16384</a>
%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 37.
%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
%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 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
%e . O O O O O O O
%e . O O O O O O
%e . O O O O O
%e . O O O O
%e . O O O
%e . O O
%e . O
%e .
%e There are 300 ON cells, so a(15) = 300.
%t 6*Join[{0}, Accumulate@ Flatten@ Table[Range[2^n, 1, -1], {n, 0, 5}]] (* _Michael De Vlieger_, Nov 03 2022 *)
%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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