A161206 V-toothpick (or honeycomb) sequence (see Comments lines for definition).

0, 1, 3, 7, 13, 21, 31, 43, 57, 69, 81, 99, 123, 153, 183, 211, 241, 261, 273, 291, 317, 351, 393, 443, 499, 553, 597, 645, 709, 791, 871, 939, 1005, 1041, 1053, 1071, 1097, 1131, 1173, 1223, 1281, 1339, 1393, 1459, 1549, 1663, 1789, 1911, 2031, 2133, 2193

Offset: 0

Author

Omar E. Pol, Jun 08 2009

Keywords

• nonn

Comments

A V-toothpick is constructed from two toothpicks of length 1 with a 120-degree angle between them, forming a V.

On the infinite hexagonal grid, we start at round 0 with no V-toothpicks.

At round 1 we place a V-toothpick anywhere in the plane.

At round 2 we place two other V-toothpicks. Note that, after round 2, in the structure there are three V-toothpicks, with seven 120-degree angles and one 240-degree angle.

At round 3 we place four other V-toothpicks.

And so on...

The structure looks like an unfinished honeycomb.

The sequence gives the number of V-toothpicks after n rounds. A161207 (the first differences) gives the number added at the n-th round.

See the entry A139250 for more information about the growth of toothpicks.

Note that, on the infinite hexagonal grid, a V-toothpick can be represented as a polyedge with two components. In this case, at n-th round, the structure is a polyedge with 2*a(n) components (or 2*a(n) toothpicks).

In the structure we can see distinct closed polygonal regions with side length equal to 1, for example: regular hexagons, concave decagons, concave dodecagons.

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.]
• Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330.
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to toothpick sequences
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A139250, A160120, A160172, A161207, A161328, A161330.

Extensions

Terms beyond a(19) from R. J. Mathar, Jan 21 2010