

A161328


Etoothpick sequence (see Comments lines for definition).


20



0, 1, 4, 9, 16, 29, 40, 57, 72, 93, 116, 141, 168, 201, 228, 253, 268, 293, 328, 369, 424, 477, 536, 597, 656, 721, 784, 841, 888, 925, 972, 1037, 1108, 1205, 1300, 1405, 1500, 1589, 1672, 1753, 1840, 1933, 2012, 2085, 2164, 2253, 2360, 2473, 2592, 2705, 2820
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OFFSET

0,3


COMMENTS

An Etoothpick is formed by three toothpicks, as an trident. The Etoothpick 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.
On the infinite triangular grid, we start at round 0 with no Etoothpicks.
At round 1 we place an Etoothpick anywhere in the plane.
At round 2 we add three more Etoothpicks.
At round 3 we add five more Etoothpicks.
And so on... (see 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. A161329 (the first differences) gives the number added at the nth round.
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. See the entry A139250 for more information about the growth of the toothpicks.
See also the snowflake sequence A161330.


LINKS

Table of n, a(n) for n=0..50.
David Applegate, The movie version
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.]
Dr. Goulu, 2012, Mayas, Kaya et curedents, Pourquoi Comment Combien blog, January 2012 (in French).
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330
N. J. A. Sloane, A single Etoothpick
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Zozoped, Illustration of the structure, a(42) = 2012 [Broken link].
Zozoped, Illustration of the structure, a(42) = 2012, "Nous avons vu se lever son étoile", Le blog du Barabel [broken link].
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata


FORMULA

For n >= 3, a(n) = 4 + Sum_{k=3..n} 2*Sum_{x=1..3} A220498(kx) + 2^((k mod 2) + 1)  7.  Christopher Hohl, Feb 24 2019


CROSSREFS

Cf. A139250, A139251, A160120, A160172, A161206, A161329, A161330.
Sequence in context: A113495 A110997 A001640 * A073141 A093175 A138992
Adjacent sequences: A161325 A161326 A161327 * A161329 A161330 A161331


KEYWORD

nonn


AUTHOR

Omar E. Pol, Jun 07 2009


EXTENSIONS

a(8) corrected, more terms appended by R. J. Mathar, Jan 21 2010
Extensive edits by Omar E. Pol, May 14 2012
I have copied the rule for adding new Etoothpicks (described by N. J. A. Sloane) from A161330.  Omar E. Pol, Dec 07 2012


STATUS

approved



