A182617 -id:A182617 - cp's OEIS frontend

A182618 Number of new grid points that are covered by the toothpicks added at n-th-stage to the toothpick spiral of A182617.

6, 4, 3, 3, 3, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3

Offset: 1

Omar E. Pol, Dec 12 2010

In the toothpick spiral the toothpicks are connected by their endpoints. See A182617 for more information.
Attempt at an explanation, R. J. Mathar, Dec 13 2010: (Start)
In the hexagonal grid, we can pick any of the hexagons as a center, and then define a ring of 6 first neighbors (hexagons adjacent to the center), then define a ring of 12 second neighbors (hexagons adjacent to any of the first ring) and so on. The current sequence describes a self-avoiding walk which starts in a spiral around the center hexagon, which covers 5 edges. The walk then takes one step to reach the rim of the first ring and travels once around this ring until it reaches a point where self-avoidance stops it. It then takes one step to reach the rim of the second ring and walks around that one, etc. Imagine that on each edge we place a toothpick if it's on the path, and interrupt counting the total number of toothpicks each time one of the hexagons has six vertices covered. The first differences of these intermediate totals define the sequence. (End)

			At stage 1, starting from a node on the hexagonal net, we place 5 toothpicks on 5 edges of the first hexagon, so a(1)= 6 because there are 6 grid points that are covered by the toothpicks.
At stage 2, starting from the last exposed endpoints, we place 4 toothpicks on the edges of the second hexagon, so a(2)=4 because there are new 4 grid points that are covered by the toothpicks.
At stage 3, starting from the last exposed endpoints we place 3 toothpicks on the edges of the third hexagon, so a(3)=3 because there are new 3 grid points covered. Etc.
If written as a triangle, begins:
6,
4,3,3,3,3,2,
3,3,2,3,2,3,2,3,2,3,2,2,
3,2,3,2,2,3,2,2,3,2,2,3,2,2,3,2,2,2,
3,2,2,3,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,3,2,2,2,2,
3,2,2,2,3,2,2,2,2,3,2,2,2,2,3,2,2,2,2,3,2,2,2,2,3,2,2,2,2,2

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
Index entries for sequences related to toothpick sequences

Cf. A121149, A139250, A139251, A182617, A182619, A182632, A182840.

Row n has A008458(n-1) terms. Row sums give A017593.

A182632 Toothpick sequence on the hexagonal net starting from a node.

Original entry on oeis.org

0, 3, 9, 21, 33, 45, 69, 105, 129, 141, 165, 213, 273, 321, 381, 465, 513, 525, 549, 597, 657, 717, 801, 933, 1065, 1137, 1197, 1317, 1485, 1629, 1785, 1977, 2073, 2085, 2109, 2157, 2217, 2277, 2361, 2493, 2625, 2709, 2793, 2949, 3177, 3405, 3633

Offset: 0

Omar E. Pol, Dec 07 2010

nonn

A connected network of toothpicks is constructed by the following iterative procedure. At stage 1, place three toothpicks each of length 1 on a hexagonal net, as a propeller, joined at a node. At each subsequent stage, add two toothpicks (which could be called a single V-toothpick with a 120-degree corner) adjacent to each node which is the endpoint of a single toothpick.
The exposed endpoints of the toothpicks of the old generation are touched by the endpoints of the toothpicks of the new generation. In the graph, the edges of the hexagons become edges of the graph, and the graph grows such that the nodes that were 1-connected in the old generation are 3-connected in the new generation.
It turns out heuristically that this growth does not show frustration, i.e., a free edge is never claimed by two adjacent exposed endpoints at the same stage; the rule of growing the network does apparently not need specifications to address such cases.
The sequence gives the number of toothpicks in the toothpick structure after n-th stage. A182633 (the first differences) gives the number of toothpicks added at n-th stage.
a(n) is also the number of components after n-th stage in a toothpick structure starting with a single Y-toothpick in stage 1 and adding only V-toothpicks in stages >= 2. For example: consider that in A161644 a V-toothpick is also a polytoothpick with two components or toothpicks and a Y-toothpick is also a polytoothpick with three components or toothpicks. For more information about this comment see A161206, A160120 and A161644.
Has a behavior similar to A151723, A182840. - Omar E. Pol, Mar 07 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.
The structure has internal growth.
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.
For another version starting with a simple toothpick see A182840.
For a version of the structure in the first quadrant but on the square grid see A182838. (End)

			a(0)=0. At stage 1 we place 3 toothpicks connected to the initial grid point of the structure. Note that there are 3 exposed endpoints. At stage 2 we place 6 toothpicks, so a(2)=3+6=9, etc.

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
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A139250, A160120, A161206, A161644, A182617, A182633, A182634, A182836, A182838, A182840.

a(n) = 3*A182634(n).

a(n) = 1 + 2*A161644(n), n >= 1. - Omar E. Pol, Mar 07 2013

A121149 Minimal number of vertices in a planar connected n-polyhex.

Original entry on oeis.org

1, 6, 10, 13, 16, 19, 22, 24, 27, 30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 54, 57, 59, 62, 64, 66, 69, 71, 73, 76, 78, 80, 83, 85, 87, 90, 92, 94, 96, 99, 101, 103, 106, 108, 110, 112, 115, 117, 119, 121, 124, 126, 128, 130, 133, 135, 137, 139, 142, 144, 146, 148, 150, 153, 155, 157, 159, 162, 164, 166, 168, 170, 173, 175, 177, 179, 181, 184, 186, 188, 190, 192, 195, 197, 199, 201, 203, 206, 208, 210, 212, 214, 216, 219, 221, 223, 225, 227, 230, 232, 234, 236

Offset: 0

Alexander Adamchuk, Aug 12 2006

a(4) appears to be wrong: the polyhex labeled "bee" on Weisstein's article has 14 vertices. - Joerg Arndt, Oct 05 2016. However, "bee" has 16 vertices when the two "interior" vertices are counted, i.e., those where three hexagons meet.  - Felix Fröhlich, Oct 05 2016
a(n) is also the size of the smallest polyhex with n disjoint holes. - Luca Petrone, Feb 28 2017
Also numbers found at the end of n-th hexagonal arc of 'graphene' number spiral (numbers in the nodes of planar net 6^3, starting with 1). See the "Illustration for the first 76 terms" link. - Yuriy Sibirmovsky, Oct 04 2016
From Ya-Ping Lu, Feb 19 2022: (Start)
For each n-polyhex (n>=3), an n-gon can be constructed by connecting the centers of external neighboring hexagons in the n-polyhex. If the n-gon is convex (n is indicated by * in the figure below), a(n+1) = a(n) + 3; otherwise, a(n+1) = a(n) + 2. For example, for n=3, triangle 1-2-3-1 is convex and a(4) = a(3) + 3 = 16. For n=17, heptagon 6-8-9-11-13-15-17-6 is nonconvex and a(18) = a(17) + 2 = 52.
.
          49--50--51--52*-53
          / \ / \ / \ / \ / \
        48*-28--29--30*-31--54
        / \ / \ / \ / \ / \ / \
      47--27*-13--14*-15--32--55
      / \ / \ / \ / \ / \ / \ / \
    46--26--12*--4*--5*-16*-33*-56*
    / \ / \ / \ / \ / \ / \ / \ / \
  45--25--11---3*--1---6--17--34--57
    \ / \ / \ / \ / \ / \ / \ / \ /
    44*-24*-10*--2---7*-18--35--58
      \ / \ / \ / \ / \ / \ / \ /
      43--23---9---8*-19*-36--59
        \ / \ / \ / \ / \ / \ /
        42--22--21*-20--37*-60
          \ / \ / \ / \ / \ /
          41--40*-39--38--61*
(End)

Moriah Elkin, Gregg Musiker, and Kayla Wright, Twists of Gr(3,n) Cluster Variables as Double and Triple Dimer Partition Functions, arXiv:2305.15531 [math.CO], 2023. See p. 18.
Luca Petrone, Illustration showing a(3) - a(43)
Yuriy Sibirmovsky, Illustration for the first 76 terms
Eric Weisstein's World of Mathematics, Polyhex.

Essentially the same as A182617: a(n) = A182617(n) + 1.

More terms from Luca Petrone, Mar 19 2017

a(0)=1 added by N. J. A. Sloane, Mar 23 2017

cp's OEIS Frontend

A182618 Number of new grid points that are covered by the toothpicks added at n-th-stage to the toothpick spiral of A182617.

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A182632 Toothpick sequence on the hexagonal net starting from a node.

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A121149 Minimal number of vertices in a planar connected n-polyhex.

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