A160157 -id:A160157 - cp's OEIS frontend

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

0, 6, 24, 42, 96, 114, 168, 222, 348, 402, 456, 510, 636, 726, 852, 1014, 1320, 1482, 1536, 1590, 1716, 1806, 1932, 2094, 2400, 2598, 2724, 2886, 3192, 3498, 3840, 4254, 4956, 5442, 5568, 5622, 5748, 5838, 5964, 6126, 6432, 6630, 6756, 6918, 7224, 7530, 7872, 8286

Offset: 0

Omar E. Pol, Jan 11 2015

nonn

Also 6 times the Y-toothpicks sequence A160120.
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.
The rules are the essentially the same as A160120.
An ON cell remains ON forever.
The sequence gives the number of triangular ON cells after the n-th stage.
A253771 (the first differences) give the number of triangular cells turned "ON" at the n-th stage.
A160120 (the Y-toothpick sequence) gives the number of stars in the structure after the n-th stage.
A160121 gives the number of stars added at the n-th stage.
A160167 gives the number of diamonds in the structure after the n-th stage.

			After one generation, the cellular automaton looks like a star or a flower with three petals as shown below:
.
.        /\
.       _\/_
.      /_/\_\
.
There are one star, three diamonds and six ON cells, so a(1) = 6.

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

a(n) = 6*A160120(n) = 3*A160157(n) = 2*A160167(n).

A160167 Total number of single toothpicks after n-th stage in the Y-toothpick structure of A160120.

Original entry on oeis.org

0, 3, 12, 21, 48, 57, 84, 111, 174, 201, 228, 255, 318, 363, 426, 507, 660, 741, 768, 795, 858, 903, 966, 1047, 1200, 1299, 1362, 1443, 1596, 1749, 1920, 2127, 2478, 2721, 2784, 2811, 2874, 2919, 2982, 3063, 3216, 3315, 3378, 3459, 3612, 3765, 3936, 4143, 4494, 4755, 4854, 4935

Offset: 0

Omar E. Pol, Jun 01 2009, Jun 09 2009

nonn

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

			From _Omar E. Pol_, Feb 10 2015: (Start)
After one generation, also, the cellular automaton looks like a star or a flower with three petals as shown below:
.
.        /\
.       _\/_
.      /_/\_\
.
There are six ON cells and three diamonds, so a(1) = 3.
(End)

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, 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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A160120, A160157, A253770.

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

a(n) = 3*A160157(n)/2 = A253770(n)/2. - Omar E. Pol, Feb 10 2015

New name and more terms from Omar E. Pol, Feb 10 2015

cp's OEIS Frontend

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

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