A161330 Snowflake (or E-toothpick) sequence (see Comments lines for definition).

0, 2, 8, 14, 20, 38, 44, 62, 80, 98, 128, 146, 176, 218, 224, 242, 260, 290, 344, 374, 452, 494, 548, 626, 668, 734, 812, 830, 872, 914, 968, 1058, 1124, 1250, 1340, 1430, 1532, 1598, 1676, 1766, 1856, 1946, 2000, 2066, 2180, 2258, 2384, 2510, 2612, 2714, 2852, 2954, 3116, 3218, 3332, 3494, 3620, 3782, 3896, 3998, 4100

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. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

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

Index entries for sequences related to cellular automata

Index entries for sequences related to toothpick sequences

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

Cf. A139250, A139251, A160120, A160172, A161206, A161328, A161331, A161333.

Sequence in context: A241003 A133229 A364615 * A046940 A046939 A082930

Adjacent sequences: A161327 A161328 A161329 * A161331 A161332 A161333

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