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

Author

Omar E. Pol, Jun 07 2009

Keywords

• nonn

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

Crossrefs

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

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]

Extensions

a(9)-a(12) from N. J. A. Sloane, Dec 07 2012

Corrected and extended by David Applegate, Dec 12 2012