A160167 - OEIS

%I #34 Feb 24 2021 02:48:18

%S 0,3,12,21,48,57,84,111,174,201,228,255,318,363,426,507,660,741,768,

%T 795,858,903,966,1047,1200,1299,1362,1443,1596,1749,1920,2127,2478,

%U 2721,2784,2811,2874,2919,2982,3063,3216,3315,3378,3459,3612,3765,3936,4143,4494,4755,4854,4935

%N Total number of single toothpicks after n-th stage in the Y-toothpick structure of A160120.

%C Also, replace the Y-toothpick with the "three-diamonds" symbol, so we have a new cellular automaton in which a(n) counts the total number of diamonds in the structure after the n-th stage, A160120 also gives the total number of "three-diamonds" symbols after the n-th stage, and A253770 gives the total number of triangular ON cells after the n-th stage. - _Omar E. Pol_, Feb 10 2015

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

%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>

%F a(n) = 3*A160120(n).

%F a(n) = 3*A160157(n)/2 = A253770(n)/2. - _Omar E. Pol_, Feb 10 2015

%e From _Omar E. Pol_, Feb 10 2015: (Start)

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

%e .

%e .        /\

%e .       _\/_

%e .      /_/\_\

%e .

%e There are six ON cells and three diamonds, so a(1) = 3.

%e (End)

%Y Cf. A139250, A160120, A160157, A253770.

%K nonn

%O 0,2

%A _Omar E. Pol_, Jun 01 2009, Jun 09 2009

%E New name and more terms from _Omar E. Pol_, Feb 10 2015