

A161330


Snowflake (or Etoothpick) sequence (see Comments lines for definition).


29



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
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OFFSET

0,2


COMMENTS

This sequence is an Etoothpick sequence (cf. A161328) but starting with two backtoback Etoothpicks.
On the infinite triangular grid, we start at round 0 with no Etoothpicks.
At round 1 we place two backtoback Etoothpicks, forming a star with six endpoints.
At round 2 we add six more Etoothpicks.
At round 3 we add six more Etoothpicks.
And so on ... (see the illustrations).
The rule for adding new Etoothpicks 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 Etoothpick 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 Etoothpicks in the structure after n rounds. A161331 (the first differences) gives the number added at the nth round.
See the entry A139250 for more information about the toothpick process and the toothpick propagation.
Note that, on the infinite triangular grid, a Etoothpick can be represented as a polyedge with three components. In this case, at nth 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 Etoothpicks.)
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.]
Ed Jeffery, Illustration of A161330 structure after 32 stages, with Etoothpicks 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 Etoothpick
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

For n >= 2, a(n) = 2 + Sum_{k=2..n} 6*A220498(k1)  6.  Christopher Hohl, Feb 24 2019


CROSSREFS

Cf. A139250, A139251, A160120, A160172, A161206, A161328, A161331, A161333.
Sequence in context: A101959 A241003 A133229 * 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



