cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 17 results. Next

A182841 Number of toothpicks added at n-th stage in the toothpick structure of A182840.

Original entry on oeis.org

0, 1, 4, 8, 14, 16, 14, 24, 38, 32, 14, 24, 46, 64, 54, 56, 86, 64, 14, 24, 46, 64, 62, 80, 126, 144, 86, 56, 110, 168, 158, 144, 198, 128, 14, 24, 46, 64, 62, 80, 126, 144, 94, 80, 142, 224, 238, 224, 286, 304, 150, 56, 110, 168, 182, 216, 326, 408, 302, 176, 262, 408, 414, 360, 438, 256
Offset: 0

Views

Author

Omar E. Pol, Dec 09 2010

Keywords

Comments

First differences of A182840.

Examples

			From _Omar E. Pol_, Nov 01 2014: (Start)
When written as an irregular triangle:
0;
1;
4;
8;
14, 16;
14, 24, 38, 32;
14, 24, 46, 64, 54, 56, 86, 64;
14, 24, 46, 64, 62, 80, 126, 144, 86, 56, 110, 168, 158, 144, 198, 128;
14, 24, 46, 64, 62, 80, 126, 144, 94, 80, 142, 224, 238, 224, 286, 304, 150, 56, 110, 168, 182, 216, 326, 408, 302, 176, 262, 408, 414, 360, 438, 256;
...
(End)

Links

Crossrefs

Cf. A139250, A139251, A160121, A161207, A182633, A182635, A151724, A182842.

Formula

a(2^k + 1) = 2^(k+2), at least for 0 <= k <= 9. - Omar E. Pol, Nov 01 2014

Extensions

More terms from Olaf Voß, Dec 24 2010
Wiki link added by Olaf Voß, Jan 14 2011

A184408 Number of holes in the toothpick structure of A182840.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 6, 6, 6, 6, 6, 8, 12, 18, 30, 30, 30, 30, 30, 30, 30, 34, 46, 56, 60, 60, 60, 66, 80, 98, 126, 126, 126, 126, 126, 126, 126, 130, 142, 150, 150, 150, 150, 154, 166, 186, 222, 248, 252, 252, 252, 252, 252, 260, 288, 322, 344, 348, 348, 364, 404, 450, 510, 510, 510, 510, 510, 510, 510, 514, 526
Offset: 0

Views

Author

Olaf Voß, Jan 13 2011

Keywords

Comments

Each connected group of unoccupied lines in the hexagonal structure enclosed by toothpicks counts as a hole.

Examples

			Structure after step 8:
      \_/
    \_/ \_/
  \_/ \_/ \_/
\_/  _/ \_  \_/
/ \_/ \_/ \_/ \
\_/ \_/ \_/ \_/
/ \ / \_/ \ / \
\_/ \_/ \_/ \_/
/ \_/ \_/ \_/ \
\_/ \_/ \_/ \_/
/ \_  \_/  _/ \
  / \_/ \_/ \
    / \_/ \
      / \
There are 6 holes in this structure, so a(8)=6.

Links

Crossrefs

For total number of missing toothpicks see A184409.

A184409 Total number of missing toothpicks in the holes in the toothpick structure of A182840.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 6, 6, 6, 6, 6, 8, 12, 22, 42, 42, 42, 42, 42, 42, 42, 46, 58, 68, 72, 72, 72, 100, 148, 150, 186, 186, 186, 186, 186, 186, 186, 190, 202, 210, 210, 210, 210, 214, 226, 254, 314, 356, 360, 360, 360, 360, 360, 368, 396, 524, 708, 652, 628, 676, 764, 710, 762, 762, 762, 762, 762, 762, 762, 766
Offset: 0

Views

Author

Olaf Voß, Jan 13 2011

Keywords

Comments

Each connected group of unoccupied lines in the hexagonal structure enclosed by toothpicks counts as a hole.

Examples

			Structure after step 8:
      \_/
    \_/ \_/
  \_/ \_/ \_/
\_/  _/ \_  \_/
/ \_/ \_/ \_/ \
\_/ \_/ \_/ \_/
/ \ / \_/ \ / \
\_/ \_/ \_/ \_/
/ \_/ \_/ \_/ \
\_/ \_/ \_/ \_/
/ \_  \_/  _/ \
  / \_/ \_/ \
    / \_/ \
      / \
There are 6 holes in this structure, in each 1 toothpick is missing, so a(8)=6.

Links

Crossrefs

For number of holes see A184408.

A175797 Size of the largest holes in the toothpick structure of A182840 after step n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 12, 12, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 48, 48, 35, 31, 27, 12, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 12, 12, 12, 7, 3, 3, 3, 3
Offset: 0

Views

Author

Olaf Voß, Jan 13 2011

Keywords

Comments

Each connected group of unoccupied lines in the hexagonal structure enclosed by toothpicks counts as a hole.

Examples

			Structure after step 8:
      \_/
    \_/ \_/
  \_/ \_/ \_/
\_/  _/ \_  \_/
/ \_/ \_/ \_/ \
\_/ \_/ \_/ \_/
/ \ / \_/ \ / \
\_/ \_/ \_/ \_/
/ \_/ \_/ \_/ \
\_/ \_/ \_/ \_/
/ \_  \_/  _/ \
  / \_/ \_/ \
    / \_/ \
      / \
There are 6 holes in this structure, in each 1 toothpick is missing (size=1), so a(8)=1.

Links

Crossrefs

For number of holes see A184408.

A178573 Number of exposed endpoints in the toothpick structure of A182840.

Original entry on oeis.org

2, 4, 8, 8, 8, 12, 20, 16, 8, 12, 24, 32, 28, 28, 44, 32, 8, 12, 24, 32, 32, 40, 64, 72, 44, 28, 56, 84, 80, 72, 100, 64, 8, 12, 24, 32, 32, 40, 64, 72, 48, 40, 72, 112, 120, 112, 144, 152, 76, 28, 56, 84
Offset: 1

Views

Author

Olaf Voß, Dec 24 2010

Keywords

Examples

			At stage 1 we place a toothpick anywhere in the plane (for example, in vertical position). There are two exposed endpoints, so a(1)=2.
At stage 2 we place 4 toothpicks: two new toothpicks touching each exposed endpoint. There are now 4 exposed endpoints, so a(2)=4.

Links

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

Views

Author

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

Keywords

Comments

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

References

  • 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).

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

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

A161644 Number of ON states after n generations of cellular automaton based on triangles.

Original entry on oeis.org

0, 1, 4, 10, 16, 22, 34, 52, 64, 70, 82, 106, 136, 160, 190, 232, 256, 262, 274, 298, 328, 358, 400, 466, 532, 568, 598, 658, 742, 814, 892, 988, 1036, 1042, 1054, 1078, 1108, 1138, 1180, 1246, 1312, 1354, 1396, 1474, 1588, 1702, 1816, 1966, 2104, 2164, 2194
Offset: 0

Views

Author

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

Keywords

Comments

Analog of A151723 and A151725, but here we are working on the hexagonal net where each triangular cell has three neighbors (meeting along its edges). A cell is turned ON if exactly one of its three neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
There is a dual version where the triangular cells meet vertex-to-vertex. The counts are the same: the two versions are isomorphic. Reed (1974) uses the vertex-to-vertex version. See the two Sloane "Illustration" links below to compare the two versions.
It appears that a(n) is also the number of polytoothpicks added in a toothpick structure formed by V-toothpicks but starting with a Y-toothpick: a(n) = a(n-1)+(A182632(n)-A182632(n-1))/2. (Checked up to n=39.) - Omar E. Pol, Dec 07 2010 and R. J. Mathar, Dec 17 2010
It appears that the behavior is similar to A161206. - Omar E. Pol, Jan 15 2016
It would be nice to have a formula or recurrence.
If new triangles are required to always move outwards we get A295559 and A295560.
From Paul Cousin, May 23 2025: (Start)
This is ETA rule 242 (11110010 in binary):
-----------------------------------------------
|state of the cell |1|1|1|1|0|0|0|0|
|sum of the neighbors' states |3|2|1|0|3|2|1|0|
|cell's next state |1|1|1|1|0|0|1|0|
----------------------------------------------- (End)

References

  • 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.]
  • S. 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 3.

Links

Crossrefs

Cf. A151723, A151725, A147562, A161206, A161645, A139250, A160120, A161206, A182632, A182840, A250300, A295559, A295560.

Programs

  • PARI
    \\ See Links section.

Formula

a(n) = (A182632(n) - 1)/2, n >= 1. - Omar E. Pol, Mar 07 2013

Extensions

Edited by N. J. A. Sloane, Jan 10 2010 and Nov 27 2017

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

Views

Author

Omar E. Pol, Dec 07 2010

Keywords

Comments

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)

Examples

			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.

Links

Crossrefs

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

Formula

a(n) = 3*A182634(n).
a(n) = 1 + 2*A161644(n), n >= 1. - Omar E. Pol, Mar 07 2013

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

Views

Author

Omar E. Pol, Dec 12 2010

Keywords

Comments

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)

Examples

			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.

Links

Crossrefs

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

Formula

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

Extensions

Partially edited by N. J. A. Sloane, Feb 06 2023
a(19)-a(41) from Omar E. Pol, Feb 06 2023

A233970 Toothpick sequence on hexagonal net starting from the vertex of a 60-degree wedge (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 51, 61, 69, 83, 99, 101, 105, 111, 121, 131, 141, 159, 183, 205, 213, 227, 249, 275, 291, 321, 353, 355, 359, 365, 375, 385, 395, 413, 437, 459, 469, 487, 515, 553, 581, 627, 683, 737, 745, 759, 781, 807
Offset: 0

Views

Author

Omar E. Pol, Dec 18 2013

Keywords

Comments

Toothpicks are connected by their endpoints. The toothpicks placed in north direction are prohibited. The sequence gives the number of toothpicks after n-th stage in the structure. A233971 (the first differences) give the number of toothpicks added at n-th stage.
First differs from A169780 at a(24).
First differs from both A233764 and A233780 at a(25).

Links

Crossrefs

Cf. A139250, A161206, A161830, A169780, A182632, A182634, A182634, A182840, A233760, A233764, A233780, A233971, A233972.
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