A182840 Toothpick sequence on hexagonal net.

0, 1, 5, 13, 27, 43, 57, 81, 119, 151, 165, 189, 235, 299, 353, 409, 495, 559, 573, 597, 643, 707, 769, 849, 975, 1119, 1205, 1261, 1371, 1539, 1697, 1841, 2039, 2167, 2181, 2205, 2251, 2315, 2377, 2457, 2583, 2727, 2821, 2901, 3043, 3267, 3505, 3729, 4015

Offset: 0

Author

Omar E. Pol, Dec 09 2010

Keywords

• nonn

Comments

Rules:

- Each new toothpick must lie on the hexagonal net such that the toothpick endpoints coincide with two consecutive nodes.

- Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of two toothpicks of new generation.

The sequence gives the number of toothpicks after n stages. A182841 (the first differences) gives the number added at the n-th stage.

The toothpick structure has polygons in which there are uncovered grid points, the same as A160120 and A161206. For more information see A139250.

Has a behavior similar to A151723, A182632. - Omar E. Pol, Feb 28 2013

From Omar E. Pol, Feb 17 2023: (Start)

Assume that every triangular cell has area 1.

It appears that the structure contains only three types of polygons:

- Regular hexagons of area 6.

- Concave decagons (or concave 10-gons) of area 12.

- Concave dodecagons (or concave 12-gons) of area 18.

There are infinitely many of these polygons.

The structure contains concentric hexagonal rings formed by hexagons and also contains concentric hexagonal rings formed by alternating decagons and dodecagons.

For an animation see the movie version in the Links section.

The animation shows the fractal-like behavior the same as in other members of the family of toothpick cellular automata.

The structure has internal growth.

For another version starting from a node see A182632.

For a version of the structure in the first quadrant but on the square grid see A182838. (End)

Examples

			We start at stage 0 with no toothpicks.
At stage 1 we place a toothpick anywhere in the plane (For example, in vertical position). There are two exposed endpoints, so a(1)=1.
At stage 2 we place 4 toothpicks. Two new toothpicks touching each exposed endpoint. So a(2)=1+4=5. There are 4 exposed endpoints.
At stage 3 we place 8 toothpicks. a(3)=5+8=13. The structure has 8 exposed endpoints.
At stage 4 we place 14 toothpicks (Not 16) because there are 4 endpoints that are touched by new 8 toothpicks but there are 4 endpoints that are touched by only 6 new toothpicks (not 8), so a(4)=13+14=27.
After 4 stages the toothpick structure has 4 hexagons and 8 exposed endpoints.
		

Links

• Olaf Voß, Table of n, a(n) for n = 0..1000
• 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.]
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Olaf Voß, Illustration of initial terms
• Index entries for sequences related to toothpick sequences
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A139250, A160120, A161206, A182632, A182634, A182838.

Extensions

More terms from Olaf Voß, Dec 24 2010

Wiki link added by Olaf Voß, Jan 14 2011