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A161328 E-toothpick sequence (see Comments lines for definition).

%I A161328 #63 Jul 27 2022 13:33:22
%S A161328 0,1,4,9,16,29,40,57,72,93,116,141,168,201,228,253,268,293,328,369,
%T A161328 424,477,536,597,656,721,784,841,888,925,972,1037,1108,1205,1300,1405,
%U A161328 1500,1589,1672,1753,1840,1933,2012,2085,2164,2253,2360,2473,2592,2705,2820
%N A161328 E-toothpick sequence (see Comments lines for definition).
%C A161328 An E-toothpick is formed by three toothpicks, as an trident. The E-toothpick has a midpoint and three exposed endpoints such that the distance between the endpoint of the central toothpick and the endpoints of the other toothpicks is equal to 1.
%C A161328 On the infinite triangular grid, we start at round 0 with no E-toothpicks.
%C A161328 At round 1 we place an E-toothpick anywhere in the plane.
%C A161328 At round 2 we add three more E-toothpicks.
%C A161328 At round 3 we add five more E-toothpicks.
%C A161328 And so on... (see illustrations).
%C A161328 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 A161328 The sequence gives the number of E-toothpicks in the structure after n rounds. A161329 (the first differences) gives the number added at the n-th round.
%C A161328 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. See the entry A139250 for more information about the growth of the toothpicks.
%C A161328 See also the snowflake sequence A161330.
%H A161328 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A161328 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 A161328 Dr. Goulu, <a href="https://www.drgoulu.com/2012/01/01/2012-mayas-kaya-et-cure-dents/#.XkzycShKjDc">2012, Mayas, Kaya et cure-dents</a>, Pourquoi Comment Combien blog, January 2012 (in French).
%H A161328 Omar E. Pol, <a href="/A161328/a161328_1.jpg">A magic wand with star in the E-toothpick cellular automaton</a>
%H A161328 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp120.jpg">Illustration of initial terms of A160120, A161206, A161328, A161330</a>
%H A161328 N. J. A. Sloane, <a href="/A161330/a161330.jpg">A single E-toothpick</a>
%H A161328 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 A161328 Zozoped, <a href="https://p.twimg.com/AiMexOHCEAAeONg.png">Illustration of the structure, a(42) = 2012</a> [Broken link].
%H A161328 Zozoped, <a href="http://blog.barabel.net/index.php?post/2012/01/epiphanie">Illustration of the structure, a(42) = 2012</a>, "Nous avons vu se lever son étoile", Le blog du Barabel [broken link].
%H A161328 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%H A161328 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A161328 For n >= 3, a(n) = 4 + Sum_{k=3..n} 2*Sum_{x=1..3} A220498(k-x) + 2^((k mod 2) + 1) - 7. - _Christopher Hohl_, Feb 24 2019
%Y A161328 Cf. A139250, A139251, A160120, A160172, A161206, A161329, A161330.
%K A161328 nonn
%O A161328 0,3
%A A161328 _Omar E. Pol_, Jun 07 2009
%E A161328 a(8) corrected, more terms appended by _R. J. Mathar_, Jan 21 2010
%E A161328 Extensive edits by _Omar E. Pol_, May 14 2012
%E A161328 I have copied the rule for adding new E-toothpicks (described by _N. J. A. Sloane_) from A161330. - _Omar E. Pol_, Dec 07 2012