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 1-7 of 7 results.

A151710 First differences of A160715.

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 9, 21, 3, 9, 9, 21, 15, 21, 27, 51, 3, 9, 9, 21, 15, 21, 27, 51, 21, 21, 27, 51
Offset: 1

Views

Author

Omar E. Pol, Jun 02 2009

Keywords

Comments

Number of Y-toothpick added at n-th stage to the structure of A160715.
Similar to A160121. [From Omar E. Pol, May 29 2010]

Examples

			Contribution from _Omar E. Pol_, Dec 18 2012 (Start):
Written as an irregular triangle begins:
1;
3;
3, 9;
3, 9, 9, 21;
3, 9, 9, 21, 15, 21, 27, 51;
3, 9, 9, 21, 15, 21, 27, 51, 21, 21, 27, 51,...
(End)

Links

Crossrefs

Cf. A000225, A006516, A139250, A160120.
Cf. A048883, A139251, A151837, A160121.

Extensions

More terms from Omar E. Pol, May 29 2010

A173068 a(n) = A160120(n) - A160715(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6, 6, 6, 6, 6, 6, 6, 30, 30, 30, 30, 30, 30, 30, 30, 42, 42, 42, 42, 42, 42, 42, 42, 120
Offset: 0

Views

Author

Omar E. Pol, May 29 2010

Keywords

Links

Crossrefs

Cf. A130665, A139250, A160120, A160715, A173066, A173067, A173451, A173452, A173453.

Extensions

a(14)-a(33) from Robert Price, Jun 17 2019

A173067 a(n) = A130665(n-1) - A160715(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 6, 6, 6, 12, 6
Offset: 1

Views

Author

Omar E. Pol, May 29 2010

Keywords

Links

Crossrefs

Cf. A130665, A139250, A160120, A160715, A173066, A173068, A173451, A173452, A173453.

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

A160121 First differences of A160120.

Original entry on oeis.org

1, 3, 3, 9, 3, 9, 9, 21, 9, 9, 9, 21, 15, 21, 27, 51, 27, 9, 9, 21, 15, 21, 27, 51, 33, 21, 27, 51, 51, 57, 69, 117, 81, 21, 9, 21, 15, 21, 27, 51, 33, 21, 27, 51, 51, 57, 69, 117, 87, 33, 27, 51, 51, 57, 75, 129, 117, 75, 69, 117, 135, 141, 171, 279, 231, 69, 9, 21, 15, 21, 27
Offset: 1

Views

Author

Omar E. Pol, May 02 2009

Keywords

Comments

Number of Y-toothpicks added at n-th stage to the Y-toothpick structure of A160120.
For a simpler version, see A151710. - Omar E. Pol, Dec 18 2012

Examples

			Contribution from _Omar E. Pol_, Jun 18 2009: (Start)
May be written as a triangle:
1,
3,
3,
9,
3,9,
9,21,9,9,
9,21,15,21,27,51,27,9,
9,21,15,21,27,51,33,21,27,51,51,57,69,117,81,21,
9,21,15,21,27,51,33,21,27,51,51,57,69,117,87,33,27,51,51,57,75,129,117,75,69,117,135,141,171,279,231,69;
Rows converge to A161326.
(End)
Contribution from _Omar E. Pol_, Dec 18 2012: (Start):
Also this sequence may be written as another triangle (according to the structure of triangle A151710):
1;
3;
3,  9;
3,  9,9,21;
9,  9,9,21,15,21,27,51;
27, 9,9,21,15,21,27,51,33,21,27,51,51,57,69,117;
81,21,9,21,15,21,27,51,33,21,27,51,51,57,69,117,87,33,27,51,51,57,75,129,117,75,69,117,135,141,171,279;
(End)

Links

Crossrefs

Cf. A139250, A139251, A151710, A160171, A160173, A160120, A160715, A161207, A161329, A161331, A161326, A161431, A147582.

Programs

  • Mathematica
    YTPFunc[lis_, step_] := With[{out = Extract[lis, {{1, 2}, {2, 1}, {-1, -1}}], in = lis[[2, 2]]}, Which[in == 1, 3, in == 0 && Count[out, 1] >= 2, 2, in == 0 && Count[out, 1] == 1, 1, True, in]]; A160121[n_] := Count[CellularAutomaton[{YTPFunc, {}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 1, 2] (* JungHwan Min, Jan 28 2016 *)
A160121[n_] := Count[CellularAutomaton[{13390417258775213635414055181254541831894674613399006361662885886563211940509571858857491972104491013971547937418035084866785430974106432144737472376143620, 4, {{-1, 0}, {0, -1}, {0, 0}, {1, 1}}}, {{{1}}, 0}, {{{n}}}], 1, 2] (* JungHwan Min, Jan 28 2016 *)

Extensions

More terms from David Applegate, Jun 14 2009

A173066 a(n) = A130665(n-1) - A160120(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 6, 6, 36, 12, 12, 12, 18, 12, 18, 18, 48, 24, 30, 30, 60, 36, 60, 72, 198, 120, 108, 108, 114, 108, 114, 114, 144, 120, 126, 126, 156, 132, 156, 168, 294, 216, 210, 210
Offset: 1

Views

Author

Omar E. Pol, May 29 2010

Keywords

Links

Crossrefs

Cf. A130665, A139250, A160120, A160715, A173067, A173068, A173451, A173452, A173453.

Extensions

More terms from Nathaniel Johnston, Nov 15 2010

A266532 Total number of Y-toothpicks after n-th stage in the "outward" version of the cellular automaton of A160120.

Original entry on oeis.org

0, 1, 4, 7, 16, 19, 28, 37, 58, 61, 70, 79, 100, 109, 130, 151, 196, 199, 208, 217, 238, 247, 268, 289, 334, 343, 364, 385, 430, 451, 496, 541, 634, 637, 646, 655, 676, 685, 706, 727, 772, 781, 802, 823, 868, 889, 934, 979, 1072, 1081, 1102, 1123, 1168, 1189, 1234, 1279, 1372, 1393, 1438, 1483, 1576, 1621, 1714, 1807, 1996, 1999, 2008, 2017
Offset: 0

Views

Author

David Applegate and Omar E. Pol, Jan 18 2016

Keywords

Comments

For the connection with A160720 (the "outward" version of the Ulam-Warburton cellular automaton A147562) see formula section and A267700.
A266533 (the first differences) gives the number of Y-toothpicks added to the structure at n-th stage.
First differs from A160120 at a(9).
First differs from A160715 at a(13).

Links

Crossrefs

Cf. A139250, A147562, A159912, A160120, A160715, A160720, A266533, A267700, A267701.

Formula

Conjecture: a(n) = 1 + 3*(A160720(n) - 1)/4 = 1 + 3*A267700(n-1), n >= 1. This formula is correct! - N. J. A. Sloane, Jan 23 2016
a(n) = 1 + 3*(A159912(n) - n)/2, n >= 1. - Omar E. Pol, Jan 24 2016
Showing 1-7 of 7 results.

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