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 1-4 of 4 results.

A253771 First differences of A253770.

Original entry on oeis.org

0, 6, 18, 18, 54, 18, 54, 54, 126, 54, 54, 54, 126, 90, 126, 162, 306, 162, 54, 54, 126, 90, 126, 162, 306, 198, 126, 162, 306, 306, 342, 414, 702, 486, 126, 54, 126, 90, 126, 162, 306, 198, 126, 162, 306, 306, 342, 414, 702, 522, 198, 162
Offset: 0

Views

Author

Omar E. Pol, Jan 13 2015

Keywords

Comments

Number of cells turned "ON" at n-th stage of cellular automaton of A253770.
Also 6 times the number of Y-toothpicks added at n-th stage in the Y-toothpick structure of A160120.

Examples

			Positive terms can be written as a triangle in which row lengths is A011782 as shown below:
6;
18;
18,  54;
18,  54,54,126;
54,  54,54,126,90,126,162,306;
162, 54,54,126,90,126,162,306,198,126,162,306,306,342,414,702;
486,126,54,126,90,126,162,306,198,126,162,306,306,342,414,702,522,198,162,...

Links

Crossrefs

Cf. A139251, A147582, A151724, A160121, A161645, A182633, A250301, A253770.

Formula

a(n) = 6*A160121(n).

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

Original entry on oeis.org

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

Views

Author

Omar E. Pol, May 02 2009

Keywords

Comments

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

Links

Crossrefs

Toothpick sequence: A139250.
Cf. A000079, A000225, A006516, A147562, A153006, A160121, A160123, A160715, A161206, A161328, A161330, A161430, A173066, A173068, A253770.

Programs

  • Mathematica
    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 *)

Extensions

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

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

Views

Author

Omar E. Pol, Jan 15 2015

Keywords

Comments

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

Links

Crossrefs

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

A160167 Total number of single toothpicks after n-th stage in the Y-toothpick structure of A160120.

Original entry on oeis.org

0, 3, 12, 21, 48, 57, 84, 111, 174, 201, 228, 255, 318, 363, 426, 507, 660, 741, 768, 795, 858, 903, 966, 1047, 1200, 1299, 1362, 1443, 1596, 1749, 1920, 2127, 2478, 2721, 2784, 2811, 2874, 2919, 2982, 3063, 3216, 3315, 3378, 3459, 3612, 3765, 3936, 4143, 4494, 4755, 4854, 4935
Offset: 0

Views

Author

Omar E. Pol, Jun 01 2009, Jun 09 2009

Keywords

Comments

Also, replace the Y-toothpick with the "three-diamonds" symbol, so we have a new cellular automaton in which a(n) counts the total number of diamonds in the structure after the n-th stage, A160120 also gives the total number of "three-diamonds" symbols after the n-th stage, and A253770 gives the total number of triangular ON cells after the n-th stage. - Omar E. Pol, Feb 10 2015

Examples

			From _Omar E. Pol_, Feb 10 2015: (Start)
After one generation, also, the cellular automaton looks like a star or a flower with three petals as shown below:
.
.        /\
.       _\/_
.      /_/\_\
.
There are six ON cells and three diamonds, so a(1) = 3.
(End)

Links

Crossrefs

Cf. A139250, A160120, A160157, A253770.

Formula

a(n) = 3*A160120(n).
a(n) = 3*A160157(n)/2 = A253770(n)/2. - Omar E. Pol, Feb 10 2015

Extensions

New name and more terms from Omar E. Pol, Feb 10 2015
Showing 1-4 of 4 results.

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