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Number of ON states after n generations of cellular automaton based on triangles, with diamonds.
4

%I #38 Mar 24 2017 00:47:57

%S 0,6,24,42,96,114,168,222,348,402,456,510,636,726,852,1014,1320,1482,

%T 1536,1590,1716,1806,1932,2094,2400,2598,2724,2886,3192,3498,3840,

%U 4254,4956,5442,5568,5622,5748,5838,5964,6126,6432,6630,6756,6918,7224,7530,7872,8286

%N Number of ON states after n generations of cellular automaton based on triangles, with diamonds.

%C Also 6 times the Y-toothpicks sequence A160120.

%C Explanation: consider the Y-toothpick structure of A160120, then replace every Y-toothpick with six ON cells forming a star with three rhombuses (or diamonds) that share only one vertex. Every diamond contains two triangular cells that share one edge.

%C The rules are the essentially the same as A160120.

%C An ON cell remains ON forever.

%C The sequence gives the number of triangular ON cells after the n-th stage.

%C A253771 (the first differences) give the number of triangular cells turned "ON" at the n-th stage.

%C A160120 (the Y-toothpick sequence) gives the number of stars in the structure after the n-th stage.

%C A160121 gives the number of stars added at the n-th stage.

%C A160167 gives the number of diamonds in the structure after the n-th stage.

%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*A160120(n) = 3*A160157(n) = 2*A160167(n).

%e After one generation, the cellular automaton looks like a star or a flower with three petals as shown below:

%e .

%e . /\

%e . _\/_

%e . /_/\_\

%e .

%e There are one star, three diamonds and six ON cells, so a(1) = 6.

%Y Cf. A139250, A147562, A151723, A160120, A160157, A160167, A161644, A182632, A250300, A253771.

%K nonn

%O 0,2

%A _Omar E. Pol_, Jan 11 2015