A256266 Total number of ON states after n generations of cellular automaton based on triangles (see Comments lines for definition).

0, 6, 18, 24, 48, 66, 78, 84, 132, 174, 210, 240, 264, 282, 294, 300, 396, 486, 570, 648, 720, 786, 846, 900, 948, 990, 1026, 1056, 1080, 1098, 1110, 1116, 1308, 1494, 1674, 1848, 2016, 2178, 2334, 2484, 2628, 2766, 2898, 3024, 3144, 3258, 3366, 3468, 3564, 3654, 3738, 3816, 3888, 3954, 4014, 4068, 4116, 4158, 4194, 4224, 4248

Offset: 0

Author

Omar E. Pol, Mar 20 2015

Keywords

• nonn
• look

Comments

On the infinite triangular grid we start at stage 0 with a hexagon formed by six OFF cells, so a(0) = 0.

At stage 1, around the mentioned hexagon, six triangular cells connected by their vertices are turned ON forming a six-pointed star, so a(1) = 6.

We use the same rules as A255748 for every one of the six 60-degree wedges of the structure.

If n is a power of 2 minus 1 and n is greater than 2, then the structure looks like concentric six-pointed stars.

If n is a power of 2 and n is greater than 2, then the structure looks like a hexagon that contains concentric six-pointed stars.

Note that in every wedge the structure seems to grow into the holes of a virtual Sierpiński's triangle (see example).

Examples

			Illustration of the structure after 15 generations:
(Note that every circle should be replaced with a triangle.)
.
.                            O
.                           O O
.                          O O O
.                         O O O O
.                        O O O O O
.                       O O O O O O
.                      O O O O O O O
.                     O O O O O O O O
.    O O O O O O O O \       O       / O O O O O O O O
.     O O O O O O O   \     O O     /   O O O O O O O
.      O O O O O O     \   O O O   /     O O O O O O
.       O O O O O       \ O O O O /       O O O O O
.        O O O O O O O O \   O   / O O O O O O O O
.         O O O   O O O   \ O O /   O O O   O O O
.          O O     O O O O \ O / O O O O     O O
.           O       O   O O \ / O O   O       O
.            - - - - - - - -   - - - - - - - -
.           O       O   O O / \ O O   O       O
.          O O     O O O O / O \ O O O O     O O
.         O O O   O O O   / O O \   O O O   O O O
.        O O O O O O O O /   O   \ O O O O O O O O
.       O O O O O       / O O O O \       O O O O O
.      O O O O O O     /   O O O   \     O O O O O O
.     O O O O O O O   /     O O     \   O O O O O O O
.    O O O O O O O O /       O       \ O O O O O O O O
.                     O O O O O O O O
.                      O O O O O O O
.                       O O O O O O
.                        O O O O O
.                         O O O O
.                          O O O
.                           O O
.                            O
.
There are 300 ON cells, so a(15) = 300.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..16384
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 37.
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A001316, A047999, A080079, A139250, A151723, A160120, A161330, A161644, A255748, A256256.

Programs

Mathematica

6*Join[{0}, Accumulate@ Flatten@ Table[Range[2^n, 1, -1], {n, 0, 5}]] (* Michael De Vlieger, Nov 03 2022 *)

Formula

a(n) = 6 * A255748(n), n >= 1.