A161330 -id:A161330 - cp's OEIS frontend

A160120 Y-toothpick sequence (see Comments lines for definition).

0, 1, 4, 7, 16, 19, 28, 37, 58, 67, 76, 85, 106, 121, 142, 169, 220, 247, 256, 265, 286, 301, 322, 349, 400, 433, 454, 481, 532, 583, 640, 709, 826, 907, 928, 937, 958, 973, 994, 1021, 1072, 1105, 1126, 1153, 1204, 1255, 1312, 1381, 1498, 1585, 1618, 1645

Offset: 0

Omar E. Pol, May 02 2009

nonn

A Y-toothpick (or Y-shaped toothpick) is formed from three toothpicks of length 1, like a star with three endpoints and only one middle-point.
On the infinite triangular grid, we start at round 0 with no Y-toothpicks.
At round 1 we place a Y-toothpick anywhere in the plane.
At round 2 we add three more Y-toothpicks. After round 2, in the structure there are three rhombuses and a hexagon.
At round 3 we add three more Y-toothpicks.
And so on ... (see illustrations).
The sequence gives the number of Y-toothpicks after n rounds. A160121 (the first differences) gives the number added at the n-th round.
The Y-toothpick pattern has a recursive, fractal (or fractal-like) structure.
Note that, on the infinite triangular grid, a Y-toothpick can be represented as a polyedge with three components. In this case, at the n-th round, the structure is a polyedge with 3*a(n) components.
This structure is more complex than the toothpick structure of A139250. For example, at some rounds we can see inward growth.
The structure contains distinct polygons which have side length equal to 1.
Observation: It appears that the region of the structure where all grid points are covered is formed only by three distinct polygons:
- Triangles
- Rhombuses
- Concave-convex hexagons
Holes in the structure: Also, we can see distinct concave-convex polygons which contains a region where there are no grid points that are covered, for example:
- Decagons   (with  1 non-covered grid point)
- Dodecagons (with  4 non-covered grid points)
- 18-gons    (with  7 non-covered grid points)
- 30-gons    (with 26 non-covered grid points)
- ...
Observation: It appears that the number of distinct polygons that contain non-covered grid points is infinite.
This sequence appears to be related to powers of 2. For example:
Conjecture: It appears that if n = 2^k, k>0, then, between the other polygons, there appears a new centered hexagon formed by three rhombuses with side length = 2^k/2 = n/2.
Conjecture: Consider the perimeter of the structure. It appears that if n = 2^k, k>0, then the structure is a triangle-shaped polygon with A000225(k)*6 sides and a half toothpick in each vertice of the "triangle".
Conjecture: It appears that if n = 2^k, k>0, then the ratio of areas between the Y-toothpick structure and the unitary triangle is equal to A006516(k)*6.
See the entry A139250 for more information about the growth of "standard" toothpicks.
See also A160715 for another version of this structure but without internal growth of Y-toothpicks. [Omar E. Pol, May 31 2010]
For an alternative visualization replace every single toothpick with a rhombus, or in other words, replace every Y-toothpick with the "three-diamond" symbol, so we have a cellular automaton in which a(n) gives the total number of "three-diamond" symbols after n-th stage and A160167(n) counts the total number of "ON" diamonds in the structure after n-th stage. See also A253770. - Omar E. Pol, Dec 24 2015
The behavior is similar to A153006 (see the graph). - Omar E. Pol, Apr 03 2018

JungHwan Min, Table of n, a(n) for n = 0..5000
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.]
David Applegate, The movie version
Ayliean McDonald, Y-toothpick fractal, Twitter video (2021).
Olbaid Fractalium, Toothpick sequence Part 2, Youtube video (2023).
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of the structure after 17 stages
Omar E. Pol, Illustration of the structure after 34 stages, from Applegate's movie version.
Omar E. Pol, Illustration: Fractal recursion, general step. (1)
Omar E. Pol, Illustration of initial terms of A139250, A160120, A147562 (Overlapping figures)
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330 (Triangular grid and toothpicks)
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

Toothpick sequence: A139250.

Cf. A000079, A000225, A006516, A147562, A153006, A160121, A160123, A160715, A161206, A161328, A161330, A161430, A173066, A173068, A253770.

YTPFunc[lis_, step_] := With[{out = Extract[lis, {{1, 2}, {2, 1}, {-1, -1}}], in = lis[[2, 2]]}, Which[in == 0 && Count[out, 2] >= 2, 1, in == 0 && Count[out, 2] == 1, 2, True, in]]; A160120[0] = 0; A160120[n_] := With[{m = n - 1}, Count[CellularAutomaton[{YTPFunc, {}, {1, 1}}, {{{2}}, 0}, {{{m}}}], 2, 2]] (* JungHwan Min, Jan 28 2016 *)
A160120[0] = 0; A160120[n_] := With[{m = n - 1}, Count[CellularAutomaton[{435225738745686506433286166261571728070, 3, {{-1, 0}, {0, -1}, {0, 0}, {1, 1}}}, {{{2}}, 0}, {{{m}}}], 2, 2]] (* JungHwan Min, Jan 28 2016 *)

More terms from David Applegate, Jun 14 2009, Jun 18 2009

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

Original entry on oeis.org

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

Omar E. Pol, Jun 08 2009

nonn

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.

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

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

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

A172310 L-toothpick sequence (see Comment lines for definition).

Original entry on oeis.org

0, 1, 3, 7, 13, 21, 33, 47, 61, 79, 97, 117, 141, 165, 203, 237, 279, 313, 339, 367, 399, 437, 489, 543, 607, 665, 733, 793, 853, 903, 969, 1039, 1109, 1183, 1233, 1285, 1345, 1399, 1463, 1529, 1613, 1701, 1817, 1923, 2055, 2155, 2291, 2417, 2557, 2663, 2781, 2881, 3003, 3109, 3247, 3361, 3499, 3631, 3783, 3939

Offset: 0

Omar E. Pol, Jan 31 2010

nonn

We define an "L-toothpick" to consist of two line segments forming an "L".
There are two size for L-toothpicks: Small and large. Each component of small L-toothpick has length 1. Each component of large L- toothpick has length sqrt(2).
The rule for the n-th stage:
If n is odd then we add the large L-toothpicks to the structure, otherwise we add the small L-toothpicks to the structure.
Note that, on the infinite square grid, every large L-toothpick is placed with angle = 45 degrees and every small L-toothpick is placed with angle = 90 degrees.
The special rule: L-toothpicks are not added if this would lead to overlap with another L-toothpick branch in the same generation.
We start at stage 0 with no L-toothpicks.
At stage 1 we place a large L-toothpick in the horizontal direction, as a "V", anywhere in the plane (Note that there are two exposed endpoints).
At stage 2 we place two small L-toothpicks.
At stage 3 we place four large L-toothpicks.
At stage 4 we place six small L-toothpicks.
And so on...
The sequence gives the number of L-toothpick after n stages. A172311 (the first differences) gives the number of L-toothpicks added at the n-th stage.
For more information see A139250, the toothpick sequence.
In calculating the extension, the "special rule" was strengthened to prohibit intersections as well as overlappings. [From John W. Layman, Feb 04 2010]
Note that the endpoints of the L-toothpicks of the new generation can touch the L-toothpìcks of old generations but the crosses and overlaps are prohibited. - Omar E. Pol, Mar 26 2016
The L-toothpick cellular automaton has an unusual property: the growths in its four wide wedges [North, East, South and West] have a recurrent behavior related to powers of 2, as we can find in other cellular automata (i.e., A194270). On the other hand, in its four narrow wedges [NE, SE, SW, NW] the behavior seems to be chaotic, without any recurrence, similar to the behavior of the snowflake cellular automaton of A161330. The remarkable fact is that with the same rules, different behaviors are produced. (See Applegate's movie version in the Links section.) - Omar E. Pol, Nov 06 2018

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
Omar E. Pol, Illustration of initial terms
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

For a similar version see A172304.
Cf. A161330 (snowflake).
Cf. A139250, A160120, A160170, A160172, A161206, A161328, A172311, A172312, A194270, A220500.

Terms a(9)-a(41) from John W. Layman, Feb 04 2010

Corrected by David Applegate and Omar E. Pol; more terms beyond a(22) from David Applegate, Mar 26 2016

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

Original entry on oeis.org

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

Omar E. Pol, Jun 07 2009

nonn

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.

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

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

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

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

A172304 L-toothpick sequence starting with two opposite L-toothpicks.

Original entry on oeis.org

0, 2, 6, 14, 22, 30, 46, 62, 70, 86, 110, 134, 166, 190, 238, 278, 302, 318, 342, 382, 430, 470, 526, 582, 646, 710, 782, 838, 902, 950, 1030, 1118, 1150, 1182, 1246, 1318, 1382, 1422, 1486, 1566, 1662, 1766, 1910, 2006, 2134, 2254, 2414, 2526, 2622

Offset: 0

Omar E. Pol, Feb 06 2010

nonn

The same as A172310 but starting with two L-toothpicks.
We start at stage 0 with no L-toothpicks.
At stage 1 we place two large L-toothpicks in the horizontal direction, as a "X", anywhere in the plane.
At stage 2 we place four small L-toothpicks.
At stage 3 we add eight more large L-toothpicks.
At stage 4 we add eight more small L-toothpicks.
And so on ...
The L-toothpick cellular automaton has an unusual property: the growths in its four wide wedges [North, East, South and West] have a recurrent behavior related to powers of 2, as we can find in other cellular automata (i.e., A212008). On the other hand, in its four narrow wedges [NE, SE, SW, NW] the behavior seems to be chaotic, without any recurrence, similar to the behavior of the snowflake cellular automaton of A161330. The remarkable fact is that with the same rules, different behaviors are produced. (See Applegate's movie version in the Links section.) - Omar E. Pol, Nov 06 2018

Yan Sheng Ang, Table of n, a(n) for n = 0..201
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

For a similar version see A172310.
Cf. A161330 (snowflake).
Cf. A139250, A160120, A160170, A160172, A161206, A161328, A172305, A172306, A172307, A172308, A172309, A172311, A172312, A172313, A212008, A220500.

Terms beyond a(14) from Yan Sheng Ang, Dec 10 2012

A222180 Total number of ON states after n generations of cellular automaton based on pentagons. Also P-toothpick sequence (see Comments lines for definition).

Original entry on oeis.org

0, 1, 6, 16, 26, 36, 56, 86, 106, 116, 136, 176, 216, 246, 296, 366, 406, 416, 436, 476, 536, 616

Offset: 0

Omar E. Pol, Mar 15 2013

Analog of A161644, A147562 and A151723, but here we are working without a lattice. Each regular pentagon has five virtual neighbors. Overlapping are prohibited. The sequence gives the number of pentagons in the structure after n-th stage. A222181 (the first differences) gives the number of pentagons added at n-th stage.

Also this is a P-toothpick sequence since every pentagon can be replaced by a P-toothpick which is formed by five toothpicks as a five-pointed star. Note that each toothpick can be represented as an apothem or as a radius of a pentagon. In both types of structures the number of toothpicks after n-th stage is equal to 5*a(n).

Omar E. Pol, Illustration of structure after 5 stages, (Contains 36 pentagons).
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A147562, A151723, A152968, A161330, A161644, A182632, A182840, A222172, A222176, A222181.

a(n) = 6 + 10*A222172(n-2), n >= 2. - Omar E. Pol, Nov 24 2013

Name improved by Omar E. Pol, Nov 24 2013

A213360 Snowflake sequence starting with six E-toothpicks.

Original entry on oeis.org

0, 6, 12, 18, 36, 42, 60, 78, 96, 126, 144, 174, 216, 222, 240, 258, 288, 342, 372, 450, 492, 546, 624, 666, 732, 810, 828, 870, 912, 966, 1056, 1122, 1248, 1338, 1428, 1530, 1596, 1674, 1764, 1854, 1944, 1998, 2064, 2178, 2256, 2382, 2508, 2610, 2712, 2850, 2952, 3114, 3216, 3330, 3492, 3618, 3780, 3894, 3996, 4098

Offset: 0

Omar E. Pol, Dec 16 2012

nonn

The structure is the same as A161330 but without the two central E-toothpicks. All terms are multiples of 6.

N. J. A. Sloane, A single E-toothpick
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A008588, A139250, A161328, A161330, A161336.

a(n) = A161330(n+1) - 2 = 6*A161336(n).

A256266 Total number of ON states after n generations of cellular automaton based on triangles (see Comments lines for definition).

Original entry on oeis.org

0, 6, 18, 24, 48, 66, 78, 84, 132, 174, 210, 240, 264, 282, 294, 300, 396, 486, 570, 648, 720, 786, 846, 900, 948, 990, 1026, 1056, 1080, 1098, 1110, 1116, 1308, 1494, 1674, 1848, 2016, 2178, 2334, 2484, 2628, 2766, 2898, 3024, 3144, 3258, 3366, 3468, 3564, 3654, 3738, 3816, 3888, 3954, 4014, 4068, 4116, 4158, 4194, 4224, 4248

Offset: 0

Omar E. Pol, Mar 20 2015

On the infinite triangular grid we start at stage 0 with a hexagon formed by six OFF cells, so a(0) = 0.
At stage 1, around the mentioned hexagon, six triangular cells connected by their vertices are turned ON forming a six-pointed star, so a(1) = 6.
We use the same rules as A255748 for every one of the six 60-degree wedges of the structure.
If n is a power of 2 minus 1 and n is greater than 2, then the structure looks like concentric six-pointed stars.
If n is a power of 2 and n is greater than 2, then the structure looks like a hexagon that contains concentric six-pointed stars.
Note that in every wedge the structure seems to grow into the holes of a virtual Sierpiński's triangle (see example).

			Illustration of the structure after 15 generations:
(Note that every circle should be replaced with a triangle.)
.
.                            O
.                           O O
.                          O O O
.                         O O O O
.                        O O O O O
.                       O O O O O O
.                      O O O O O O O
.                     O O O O O O O O
.    O O O O O O O O \       O       / O O O O O O O O
.     O O O O O O O   \     O O     /   O O O O O O O
.      O O O O O O     \   O O O   /     O O O O O O
.       O O O O O       \ O O O O /       O O O O O
.        O O O O O O O O \   O   / O O O O O O O O
.         O O O   O O O   \ O O /   O O O   O O O
.          O O     O O O O \ O / O O O O     O O
.           O       O   O O \ / O O   O       O
.            - - - - - - - -   - - - - - - - -
.           O       O   O O / \ O O   O       O
.          O O     O O O O / O \ O O O O     O O
.         O O O   O O O   / O O \   O O O   O O O
.        O O O O O O O O /   O   \ O O O O O O O O
.       O O O O O       / O O O O \       O O O O O
.      O O O O O O     /   O O O   \     O O O O O O
.     O O O O O O O   /     O O     \   O O O O O O O
.    O O O O O O O O /       O       \ O O O O O O O O
.                     O O O O O O O O
.                      O O O O O O O
.                       O O O O O O
.                        O O O O O
.                         O O O O
.                          O O O
.                           O O
.                            O
.
There are 300 ON cells, so a(15) = 300.

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 37.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A001316, A047999, A080079, A139250, A151723, A160120, A161330, A161644, A255748, A256256.

6*Join[{0}, Accumulate@ Flatten@ Table[Range[2^n, 1, -1], {n, 0, 5}]] (* Michael De Vlieger, Nov 03 2022 *)

a(n) = 6 * A255748(n), n >= 1.

A211976 First differences of the E-toothpick sequence A211964.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 4, 1, 2, 2, 3, 5, 3, 7, 4, 5, 7, 4, 6, 7, 2, 4, 4, 5, 8, 6, 11, 8, 8, 9, 6, 7, 8, 8, 8, 5, 6, 10, 7, 11, 11, 9, 9, 12, 9, 14, 9, 10, 14, 11, 14, 10, 9, 9, 11, 16, 16, 13, 12, 11, 14, 14, 16, 17, 10, 10, 14

Offset: 1

Omar E. Pol, Dec 15 2012

nonn

Number of E-toothpicks added at n-th stage to the triangular structure of A211964.

Cf. A139250, A139251, A161330, A211964.

a(n) = ((A161331(n+1)/6) + 1)/2.

A220498 Number of E-toothpicks (or tridents) added at n-th stage to the structure of the equilateral triangle of A220478.

Original entry on oeis.org

0, 2, 2, 2, 4, 2, 4, 4, 4, 6, 4, 6, 8, 2, 4, 4, 6, 10, 6, 14, 8, 10, 14, 8, 12, 14, 4, 8, 8, 10, 16, 12, 22, 16, 16, 18, 12, 14, 16, 16, 16, 10, 12, 20, 14, 22, 22, 18, 18, 24, 18, 28, 18, 20, 28, 22, 28, 20, 18, 18, 22, 32, 32, 26, 24, 22, 28, 28, 32, 34, 20, 20, 28

Offset: 0

Omar E. Pol, Feb 19 2013

nonn

Essentially the first differences of A220478.

Cf. A139250, A139251, A160121, A161330, A161329, A161331, A211974, A211976.

a(n) = 1 + A161331(n+1)/6 = 2*A211976(n).

cp's OEIS Frontend

A160120 Y-toothpick sequence (see Comments lines for definition).

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

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A172310 L-toothpick sequence (see Comment lines for definition).

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

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A172304 L-toothpick sequence starting with two opposite L-toothpicks.

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A222180 Total number of ON states after n generations of cellular automaton based on pentagons. Also P-toothpick sequence (see Comments lines for definition).

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A213360 Snowflake sequence starting with six E-toothpicks.

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A256266 Total number of ON states after n generations of cellular automaton based on triangles (see Comments lines for definition).

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A211976 First differences of the E-toothpick sequence A211964.

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