A253771 -id:A253771 - 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).

A250301 First differences of A250300.

Original entry on oeis.org

0, 3, 3, 6, 12, 12, 6, 12, 18, 18, 18, 18, 36, 36, 12, 24, 30, 18, 18, 24, 42, 54, 42, 36, 60, 66, 54, 48, 90, 90, 30, 54, 54, 18, 18, 24, 42, 54, 42, 42, 66, 78, 78, 78, 114, 150, 102, 72, 108

Offset: 0

Omar E. Pol, Jan 17 2015

Number of cells turned "ON" at n-th stage of cellular automaton of A250300.

Compare A161645.

			Positive terms can be written as a triangle in which row lengths is A011782 as shown below:
3;
3;
6,  12;
12,  6, 12, 18;
18, 18, 18, 36, 36, 12, 24, 30;
18, 18, 24, 42, 54, 42, 36, 60, 66, 54, 48, 90, 90, 30, 54, 54;
18, 18, 24, 42, 54, 42, 42, 66, 78, 78, 78, 114, 150, 102, 72, 108, ...

Cf. A139251, A147582, A151724, A160121, A161645, A182633, A250300, A253771.

cp's OEIS Frontend

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

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