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A253770 Number of ON states after n generations of cellular automaton based on triangles, with diamonds.

%I A253770 #38 Mar 24 2017 00:47:57
%S A253770 0,6,24,42,96,114,168,222,348,402,456,510,636,726,852,1014,1320,1482,
%T A253770 1536,1590,1716,1806,1932,2094,2400,2598,2724,2886,3192,3498,3840,
%U A253770 4254,4956,5442,5568,5622,5748,5838,5964,6126,6432,6630,6756,6918,7224,7530,7872,8286
%N A253770 Number of ON states after n generations of cellular automaton based on triangles, with diamonds.
%C A253770 Also 6 times the Y-toothpicks sequence A160120.
%C A253770 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 A253770 The rules are the essentially the same as A160120.
%C A253770 An ON cell remains ON forever.
%C A253770 The sequence gives the number of triangular ON cells after the n-th stage.
%C A253770 A253771 (the first differences) give the number of triangular cells turned "ON" at the n-th stage.
%C A253770 A160120 (the Y-toothpick sequence) gives the number of stars in the structure after the n-th stage.
%C A253770 A160121 gives the number of stars added at the n-th stage.
%C A253770 A160167 gives the number of diamonds in the structure after the n-th stage.
%H A253770 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 A253770 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A253770 a(n) = 6*A160120(n) = 3*A160157(n) = 2*A160167(n).
%e A253770 After one generation, the cellular automaton looks like a star or a flower with three petals as shown below:
%e A253770 .
%e A253770 .        /\
%e A253770 .       _\/_
%e A253770 .      /_/\_\
%e A253770 .
%e A253770 There are one star, three diamonds and six ON cells, so a(1) = 6.
%Y A253770 Cf. A139250, A147562, A151723, A160120, A160157, A160167, A161644, A182632, A250300, A253771.
%K A253770 nonn
%O A253770 0,2
%A A253770 _Omar E. Pol_, Jan 11 2015