OFFSET
0,2
COMMENTS
This sequence is an E-toothpick sequence (cf. A161328) but starting with two back-to-back E-toothpicks.
On the infinite triangular grid, we start at round 0 with no E-toothpicks.
At round 1 we place two back-to-back E-toothpicks, forming a star with six endpoints.
At round 2 we add six more E-toothpicks.
At round 3 we add six more E-toothpicks.
And so on ... (see the illustrations).
The rule for adding new E-toothpicks is as follows. Each E has three ends, which initially are free. If the ends of two E's meet, those ends are no longer free. To go from round n to round n+1, we add an E-toothpick at each free end (extending that end in the direction it is pointing), subject to the condition that no end of any new E can touch any end of an existing E from round n or earlier. (Two new E's are allowed to touch.)
The sequence gives the number of E-toothpicks in the structure after n rounds. A161331 (the first differences) gives the number added at the n-th round.
See the entry A139250 for more information about the toothpick process and the toothpick propagation.
Note that, on the infinite triangular grid, a E-toothpick can be represented as a polyedge with three components. In this case, at n-th round, the structure is a polyedge with 3*a(n) components.
LINKS
David Applegate, Table of n, a(n) for n = 0..1000
David Applegate, The movie version
David Applegate, Illustration of structure after 32 stages. (Contains 1124 E-toothpicks.)
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
Ed Jeffery, Illustration of A161330 structure after 32 stages, with E-toothpicks replace by rhombi (the figure on the right is the complementary structure)
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330 (Triangular grid and toothpicks) [From Omar E. Pol, Dec 06 2009]
N. J. A. Sloane, A single E-toothpick
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
FORMULA
[No formula or recurrence is known, - N. J. A. Sloane, Oct 13 2023]
For n >= 2, a(n) = 2 + Sum_{k=2..n} 6*A220498(k-1) - 6. - Christopher Hohl, Feb 24 2019. [This is a restatement of the definition. - N. J. A. Sloane, Oct 13 2023]
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jun 07 2009
EXTENSIONS
a(9)-a(12) from N. J. A. Sloane, Dec 07 2012
Corrected and extended by David Applegate, Dec 12 2012
STATUS
approved
