OFFSET
0,2
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).
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
FORMULA
a(n) = 6 * A255748(n), n >= 1.
EXAMPLE
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.
MATHEMATICA
6*Join[{0}, Accumulate@ Flatten@ Table[Range[2^n, 1, -1], {n, 0, 5}]] (* Michael De Vlieger, Nov 03 2022 *)
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Omar E. Pol, Mar 20 2015
STATUS
approved