A161644 -id:A161644 - cp's OEIS frontend

A161645 First differences of A161644: number of new ON cells at generation n of the triangular cellular automaton described in A161644.

0, 1, 3, 6, 6, 6, 12, 18, 12, 6, 12, 24, 30, 24, 30, 42, 24, 6, 12, 24, 30, 30, 42, 66, 66, 36, 30, 60, 84, 72, 78, 96, 48, 6, 12, 24, 30, 30, 42, 66, 66, 42, 42, 78, 114, 114, 114, 150, 138, 60, 30, 60, 84, 90, 114, 174, 198, 132, 90, 144, 210, 192, 192, 210, 96, 6, 12, 24

Offset: 0

David Applegate and N. J. A. Sloane, Jun 15 2009

See the comments in A161644.

It appears that a(n) is also the number of V-toothpicks or Y-toothpicks added at the n-th stage in a toothpick structure on hexagonal net, starting with a single Y-toothpick in stage 1 and adding only V-toothpicks in stages >=2 (see A161206, A160120, A182633). - Omar E. Pol, Dec 07 2010

			From _Omar E. Pol_, Apr 08 2015: (Start)
The positive terms written as an irregular triangle in which the row lengths are the terms of A011782:
1;
3;
6,6;
6,12,18,12;
6,12,24,30,24,30,42,24;
6,12,24,30,30,42,66,66,36,30,60,84,72,78,96,48;
6,12,24,30,30,42,66,66,42,42,78,114,114,114,150,138,60,30,60,84,90,114,174,198,132,90,144,210,192,192,210,96;
...
It appears that the right border gives A003945.
(End)

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Describes the dual structure where new triangles are joined at vertices rather than edges.]

Rémy Sigrist, Table of n, a(n) for n = 0..10000
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.]
R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
N. J. A. Sloane, Illustration of first 7 generations of A161644 and A295560 (edge-to-edge version)
N. J. A. Sloane, Illustration of first 11 generations of A161644 and A295560 (vertex-to-vertex version) [Include the 6 cells marked x to get A161644(11), exclude them to get A295560(11).]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A161644, A139251, A160161, A161207, A182633, A295559, A295560.

See A342271 for a(n)/3.

A170883 "Corner" sequence associated with A161644 and A161645.

Original entry on oeis.org

0, 1, 3, 7, 12, 17, 24, 35, 46, 53, 60, 73, 92, 111, 130, 155, 178, 189, 196, 209, 228, 249, 274, 311, 356, 391, 414, 445, 494, 547, 598, 657, 706, 725, 732, 745, 764, 785, 810, 847, 892, 929, 958, 999, 1064, 1141, 1218, 1307, 1404, 1471, 1502

Offset: 0

David Applegate and N. J. A. Sloane, Jan 08 2010

nonn

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 0..1000
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
David Applegate, The movie version of A139250, etc. [See A161644, corner version.]

Cf. A139250, A161644, A161645, A170882.

A295560 Same as A161644 except that triangles must always grow outwards.

Original entry on oeis.org

0, 1, 4, 10, 16, 22, 34, 52, 64, 70, 82, 100, 118, 136, 166, 208, 232, 238, 250, 268, 286, 304, 334, 376, 406, 424, 454, 496, 538, 580, 646, 736, 784, 790, 802, 820, 838, 856, 886, 928, 958, 976, 1006, 1048, 1090, 1132, 1198, 1288, 1342, 1360, 1390, 1432, 1474

Offset: 0

N. J. A. Sloane, Nov 27 2017

nonn

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Describes the dual structure where new triangles are joined at vertices rather than edges.]

Lars Blomberg, Table of n, a(n) for n = 0..10000
R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
N. J. A. Sloane, Illustration of first 7 generations of A161644 and A295560 (edge-to-edge version)
N. J. A. Sloane, Illustration of first 11 generations of A161644 and A295560 (vertex-to-vertex version) [Include the 6 cells marked x to get A161644(11), exclude them to get A295560(11).]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A161644, A161645.

Partial sums of A295559.

Terms a(18) and beyond from Lars Blomberg, Dec 20 2017

A250300 Number of ON states after n generations of cellular automaton based on triangles (compare A161644).

Original entry on oeis.org

0, 3, 6, 12, 24, 36, 42, 54, 72, 90, 108, 126, 162, 198, 210, 234, 264, 282, 300, 324, 366, 420, 462, 498, 558, 624, 678, 726, 816, 906, 936, 990, 1044, 1062, 1080, 1104, 1146, 1200, 1242, 1284, 1350, 1428, 1506, 1584, 1698, 1848, 1950, 2022, 2130

Offset: 0

Omar E. Pol, Jan 15 2015

nonn

The same rules as A161644 but here we start with three ON cells which share only one vertex.

Cf. A139250, A147562, A151723, A160120, A161644, A182632, A253770.

A170882 First differences of "corner" sequence associated with A161644 and A161645.

Original entry on oeis.org

0, 1, 2, 4, 5, 5, 7, 11, 11, 7, 7, 13, 19, 19, 19, 25, 23, 11, 7, 13, 19, 21, 25, 37, 45, 35, 23, 31, 49, 53, 51, 59, 49, 19, 7, 13, 19, 21, 25, 37, 45, 37, 29, 41, 65, 77, 77, 89, 97, 67, 31, 31, 49, 59, 69, 97, 125, 111, 75, 79, 119, 135, 129, 135, 103, 35, 7

Offset: 0

David Applegate and N. J. A. Sloane, Jan 08 2010

nonn

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 0..1000
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
David Applegate, The movie version of A139250, etc. [See A161644, corner version.]

Cf. A139250, A161644, A161645, A170883.

A151723 Total number of ON states after n generations of cellular automaton based on hexagons.

Original entry on oeis.org

0, 1, 7, 13, 31, 37, 55, 85, 127, 133, 151, 181, 235, 289, 331, 409, 499, 505, 523, 553, 607, 661, 715, 817, 967, 1069, 1111, 1189, 1327, 1489, 1603, 1789, 1975, 1981, 1999, 2029, 2083, 2137, 2191, 2293, 2443, 2545, 2599, 2701, 2875, 3097, 3295

Offset: 0

David Applegate and N. J. A. Sloane, Jun 13 2009

nonn

Analog of A151725, but here we are working on the triangular lattice (or the A_2 lattice) where each hexagonal cell has six neighbors.
A cell is turned ON if exactly one of its six neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
It would be nice to find a recurrence for this sequence!
Has a behavior similar to A182840 and possibly to A182632. - Omar E. Pol, Jan 15 2016

S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 (see Example 6, page 224).

N. J. A. Sloane, Table of n, a(n) for n = 0..4095 [First 1026 terms from David Applegate and N. J. A. Sloane]
David Applegate, The movie version
David Applegate and N. J. A. Sloane, Table of n, A151724(n), A151723(n) for n = 0..1025
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.]
Bradley Klee, Log-periodic coloring, over the half-hexagon tiling.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021

Cf. A147562, A151724, A151725, A161206, A161644, A169779, A169780, A170898, A170905, A182632, A182840, A256536, A256537.

A151723[0] = 0; A151723[n_] := Total[CellularAutomaton[{10926, {2, {{2, 2, 0}, {2, 1, 2}, {0, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{{n - 1}}}], 2]; Array[A151723, 47, 0](* JungHwan Min, Sep 01 2016 *)
A151723L[n_] := Prepend[Total[#, 2] & /@ CellularAutomaton[{10926, {2, {{2, 2, 0}, {2, 1, 2}, {0, 2, 2}}}, {1, 1}}, {{{1}}, 0}, n - 1], 0]; A151723L[46] (* JungHwan Min, Sep 01 2016 *)

a(n) = 6*A169780(n) - 6*n + 1 (this is simply the definition of A169780).
a(n) = 1 + 6*A169779(n-2), n >= 2. - Omar E. Pol, Mar 19 2015
It appears that a(n) = a(n-2) + 3*(A256537(n) - 1), n >= 3. - Omar E. Pol, Apr 04 2015

Edited by N. J. A. Sloane, Jan 10 2010

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

A182633 Number of toothpicks added at n-th stage in the toothpick structure of A182632.

Original entry on oeis.org

0, 3, 6, 12, 12, 12, 24, 36, 24, 12, 24, 48, 60, 48, 60, 84, 48, 12, 24, 48, 60, 60, 84, 132, 132, 72, 60, 120, 168, 144, 156, 192, 96, 12, 24, 48, 60, 60, 84, 132, 132, 84, 84, 156, 228, 228, 228

Offset: 0

Omar E. Pol, Dec 07 2010

First differences of A182632.

a(n) is also the number of components added at n-th stage in the toothpick structure formed by V-toothpicks with an initial Y-toothpick, since a V-toothpick has two components and a Y-toothpick has three components (For more information see A161206, A160120, A161644).

			From _Omar E. Pol_, Feb 08 2013 (Start):
When written as a triangle:
0;
3;
6;
12,12;
12,24,36,24;
12,24,48,60,48,60, 84, 48;
12,24,48,60,60,84,132,132,72,60,120,168,144,156,192,96;
12,24,48,60,60,84,132,132,84,84,156,228,228,228,...
...
It appears that positive terms of the right border are A007283.
(End)

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

A139250, A139251, A160120, A160121, A161206, A161207, A161644, A161645, A182632.

It appears that a(n) = 2*A161645(n) but with a(1)=3.

a(n) = 3*A182635(n). - Omar E. Pol, Feb 09 2013

A182634 Toothpick sequence on hexagonal net starting at the vertex of an infinite 120-degree wedge.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 71, 91, 107, 127, 155, 171, 175, 183, 199, 219, 239, 267, 311, 355, 379, 399, 439, 495, 543, 595, 659, 691, 695, 703, 719, 739, 759, 787, 831, 875, 903, 931, 983, 1059, 1135, 1211

Offset: 0

Omar E. Pol, Dec 08 2010

The sequence gives the number of toothpicks after n stages. A182635 (the first differences) gives the number added at the n-th stage.
The 120-degree wedge defines a conic region which toothpicks (except one end point of the initial toothpick) are not allowed to cross or touch. The wings of the wedge point +-60 degrees away from the pointing direction of the initial toothpick.
Toothpicks are connected by their endpoints, the same as the toothpicks of A182632.
First differs from A139250 at a(11).

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 [From _Omar E. Pol_, Dec 06 2009]
Index entries for sequences related to cellular automata

Cf. A139250, A160406, A160120, A161206, A161644, A182632, A182633, A182635.

a(n) = A182632(n)/3.

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

cp's OEIS Frontend

A161645 First differences of A161644: number of new ON cells at generation n of the triangular cellular automaton described in A161644.

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A170883 "Corner" sequence associated with A161644 and A161645.

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A295560 Same as A161644 except that triangles must always grow outwards.

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A250300 Number of ON states after n generations of cellular automaton based on triangles (compare A161644).

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A170882 First differences of "corner" sequence associated with A161644 and A161645.

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A151723 Total number of ON states after n generations of cellular automaton based on hexagons.

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

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A182633 Number of toothpicks added at n-th stage in the toothpick structure of A182632.

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A182634 Toothpick sequence on hexagonal net starting at the vertex of an infinite 120-degree wedge.

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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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