%I #52 Feb 09 2024 09:04:31
%S 0,2,8,14,20,38,44,62,80,98,128,146,176,218,224,242,260,290,344,374,
%T 452,494,548,626,668,734,812,830,872,914,968,1058,1124,1250,1340,1430,
%U 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
%N Snowflake (or E-toothpick) sequence (see Comments lines for definition).
%C This sequence is an E-toothpick sequence (cf. A161328) but starting with two back-to-back E-toothpicks.
%C On the infinite triangular grid, we start at round 0 with no E-toothpicks.
%C At round 1 we place two back-to-back E-toothpicks, forming a star with six endpoints.
%C At round 2 we add six more E-toothpicks.
%C At round 3 we add six more E-toothpicks.
%C And so on ... (see the illustrations).
%C 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.)
%C 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.
%C See the entry A139250 for more information about the toothpick process and the toothpick propagation.
%C 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.
%H David Applegate, <a href="/A161330/b161330.txt">Table of n, a(n) for n = 0..1000</a>
%H David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H David Applegate, <a href="/A161330/a161330.png">Illustration of structure after 32 stages.</a> (Contains 1124 E-toothpicks.)
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]
%H Ed Jeffery, <a href="/A161330/a161330.pdf">Illustration of A161330 structure after 32 stages, with E-toothpicks replace by rhombi</a> (the figure on the right is the complementary structure)
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp120.jpg">Illustration of initial terms of A160120, A161206, A161328, A161330 (Triangular grid and toothpicks)</a> [From _Omar E. Pol_, Dec 06 2009]
%H N. J. A. Sloane, <a href="/A161330/a161330.jpg">A single E-toothpick</a>
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F [No formula or recurrence is known, - _N. J. A. Sloane_, Oct 13 2023]
%F 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]
%Y Cf. A139250, A139251, A160120, A160172, A161206, A161328, A161331, A161333.
%K nonn
%O 0,2
%A _Omar E. Pol_, Jun 07 2009
%E a(9)-a(12) from _N. J. A. Sloane_, Dec 07 2012
%E Corrected and extended by _David Applegate_, Dec 12 2012