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

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

Offset: 0

Author

Omar E. Pol, Jun 07 2009

Keywords

• nonn

Comments

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.

On the infinite triangular grid, we start at round 0 with no E-toothpicks.

At round 1 we place an E-toothpick anywhere in the plane.

At round 2 we add three more E-toothpicks.

At round 3 we add five more E-toothpicks.

And so on... (see 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. A161329 (the first differences) gives the number added at the n-th round.

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.

See also the snowflake sequence A161330.

Links

• 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), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
• Dr. Goulu, 2012, Mayas, Kaya et cure-dents, Pourquoi Comment Combien blog, January 2012 (in French).
• Omar E. Pol, A magic wand with star in the E-toothpick cellular automaton
• Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330
• N. J. A. Sloane, A single E-toothpick
• 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

Crossrefs

Cf. A139250, A139251, A160120, A160172, A161206, A161329, A161330.

Formula

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

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 E-toothpicks (described by N. J. A. Sloane) from A161330. - Omar E. Pol, Dec 07 2012