A172308 -id:A172308 - cp's OEIS frontend

A172309 Number of L-toothpicks added to the L-toothpick structure of A172308 (First quadrant) at the n-th stage.

0, 1, 2, 2, 2, 4, 4, 2, 4, 6, 6, 8, 6, 12, 10, 6, 4, 6, 10, 12, 10, 14, 14, 16, 16, 18, 14, 16, 12, 20, 22, 8, 8, 16, 18, 16, 10, 16, 20, 24, 26, 36, 24, 32, 30, 40, 28, 24

Offset: 0

Omar E. Pol, Feb 06 2010

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

Cf. First differences of A172308.

Cf. A139250, A139251, A152978, A172304, A172305, A172306, A172307, A172308, A172310, A172311, A172312, A172313.

a(0) = 0, a(n) = A172305(n+1)/4, for n>=1.

More terms from Colin Barker, Apr 19 2015

A182838 H-toothpick sequence in the first quadrant starting with a D-toothpick placed on the diagonal [(0,1), (1,2)] (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 21, 31, 39, 43, 49, 61, 77, 91, 105, 127, 143, 147, 153, 165, 181, 197, 217, 249, 285, 307, 321, 349, 391, 431, 467, 517, 549, 553, 559, 571, 587, 603, 623, 655, 691, 715

Offset: 0

Omar E. Pol, Dec 12 2010

An H-toothpick sequence is a toothpick sequence on a square grid that resembles a partial honeycomb of hexagons.
The structure has two types of elements: the classic toothpicks with length 1 and the "D-toothpicks"  with length sqrt(2).
Classic toothpicks are placed in the vertical direction and D-toothpicks are placed in a diagonal direction.
Each hexagon has area = 4.
The network looks like an elongated hexagonal lattice placed on the square grid so that all nodes of the hexagonal net coincide with some of the grid points of the square grid. Each node in the hexagonal network is represented with coordinates x,y.
The sequence gives the number of toothpicks and D-toothpicks after n steps. A182839 (first differences) gives the number added at the n-th stage.
[It appears that for this sequence a classic toothpick is a line segment of length 1 that is parallel to the y-axis. A D-toothpick is a line segment of length sqrt(2) with slope +-1. D stands for diagonal. It also appears that classic toothpicks are not placed on the y-axis. - N. J. A. Sloane, Feb 06 2023]
From Omar E. Pol, Feb 17 2023: (Start)
This cellular automaton appears to be a version on the square grid of the first quadrant of the structure of A182840.
The rules are as follows:
- The elements (toothpicks and D-toothpicks) are connected at their ends.
- At each free end of the elements of the old generation two elements of the new generation must be connected.
- The toothpicks of length 1 must always be placed vertically, i.e. parallel to the Y-axis.
- The angle between a toothpick of length 1 and a D-toothpick of length sqrt(2) that share the same node must be 135 degrees, therefore the angle between two D-toothpicks that share the same node is 90 degrees.
As a result of these rules we can see that in the odd-indexed rows of the structure are placed only the toothpicks of length 1 and in the even-indexed rows of the structure are placed the D-toothpicks of length sqrt(2).
Apart from the trapezoids, pentagons and heptagons that are adjacent to the axes of the first quadrant it appears that there are only three types of polygons:
- Regular hexagons of area 4.
- Concave decagons (or concave 10-gons) of area 8.
- Concave dodecagons (or concave 12-gons) of area 12.
There are infinitely many of these polygons.
The structure shows a fractal-like behavior as we can see in other members of the family of toothpick cellular automata.
The structure has internal growth as some members of the mentioned family. (End)

			We start at stage 0 with no toothpicks.
At stage 1 we place a D-toothpick [(0,1),(1,2)], so a(1)=1.
At stage 2 we place a toothpick [(1,2),(1,3)] and a D-toothpick [(1,2),(2,1)], so a(2)=1+2=3.
At stage 3 we place 4 elements: a D-toothpick [(1,3),(0,4)], a D-toothpick [(1,3),(2,4)], a D-toothpick [(2,1),(3,2)] and a toothpick [(2,1),(2,0)], so a(3)=3+4=7. Etc.
The first hexagon appears in the structure after 4 stages.

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
Omar E. Pol, Illustration of a(32) = 549
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A153000, A161206, A170888, A172308, A182632, A182634, A182839, A182840, A187212, A194444, A220524, A233970, A267458, A267694, A267698.

See A360501 and A360512 for another hexagonal net built on the square grid. - N. J. A. Sloane, Feb 09 2023

Conjecture: a(n) = (A182840(n+1) + A267458(n+1) - 2)/4. - Omar E. Pol, Feb 10 2023

Partially edited by N. J. A. Sloane, Feb 06 2023

a(19)-a(41) from Omar E. Pol, Feb 06 2023

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

A172305 Number of L-toothpicks added to the L-toothpick structure of A172304 at the n-th stage.

Original entry on oeis.org

0, 2, 4, 8, 8, 8, 16, 16, 8, 16, 24, 24, 32, 24, 48, 40, 24, 16, 24, 40, 48, 40, 56, 56, 64, 64, 72, 56, 64, 48, 80, 88, 32, 32, 64, 72, 64, 40, 64, 80, 96, 104, 144, 96, 128, 120, 160, 112, 96, 80, 128, 128, 144, 88, 136, 128, 136, 144, 168, 168, 216, 160, 192, 168, 64, 80, 128

Offset: 0

Omar E. Pol, Feb 06 2010

nonn

Robert Price, Table of n, a(n) for n = 0..202
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

Cf. First differences of A172304.

Cf. A139250, A139251, A172304, A172306, A172307, A172308, A172309, A172310, A172311, A172312, A172313.

More terms from Colin Barker, Apr 19 2015

a(49)-a(66) from Robert Price, Jun 17 2019

A172306 a(n) = A172304(n)/2.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 31, 35, 43, 55, 67, 83, 95, 119, 139, 151, 159, 171, 191, 215, 235, 263, 291, 323, 355, 391, 419, 451, 475, 515, 559, 575, 591, 623, 659, 691, 711, 743, 783, 831, 883, 955, 1003, 1067, 1127, 1207, 1263, 1311, 1351, 1415, 1479, 1551, 1595, 1663, 1727, 1795, 1867, 1951, 2035, 2143, 2223, 2319, 2403, 2435, 2475, 2539, 2587

Offset: 0

Omar E. Pol, Feb 06 2010

nonn

Robert Price, Table of n, a(n) for n = 0..202
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

Cf. A139250, A172304, A172305, A172307, A172308, A172309, A172310, A172311, A172312, A172313.

a(14)-a(67) from Robert Price, Jun 17 2019

A172307 a(n) = A172305(n)/2.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 8, 8, 4, 8, 12, 12, 16, 12, 24, 20, 12, 8, 12, 20, 24, 20, 28, 28, 32, 32, 36, 28, 32, 24, 40, 44, 16, 16, 32, 36, 32, 20, 32, 40, 48, 52, 72, 48, 64, 60, 80, 56, 48, 40, 64, 64, 72, 44, 68, 64, 68, 72, 84, 84, 108, 80, 96, 84, 32, 40, 64

Offset: 0

Omar E. Pol, Feb 06 2010

nonn

Robert Price, Table of n, a(n) for n = 0..202
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

Cf. A139250, A172304, A172305, A172306, A172308, A172309, A172310, A172311, A172312, A172313.

a(14)-a(66) from Robert Price, Jun 17 2019

cp's OEIS Frontend

A172309 Number of L-toothpicks added to the L-toothpick structure of A172308 (First quadrant) at the n-th stage.

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A182838 H-toothpick sequence in the first quadrant starting with a D-toothpick placed on the diagonal [(0,1), (1,2)] (see Comments for precise definition).

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

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A172305 Number of L-toothpicks added to the L-toothpick structure of A172304 at the n-th stage.

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A172306 a(n) = A172304(n)/2.

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A172307 a(n) = A172305(n)/2.

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A172313 a(n) = A172309(n+1)/2.

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