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 10 results.

A342236 a(n) is the smallest m such that a regular m-gon with all diagonals drawn contains a cell with n sides, as in A342222, but for odd m the central m-sided polygon is not considered. Otherwise a(n) = -1 if no such m exists.

Original entry on oeis.org

4, 6, 7, 9, 15, 13, 35, 29, 29, 40, 93, 43, 399, 212
Offset: 3

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 06 2021

Keywords

Comments

An m-gon with an odd number of sides contains a central cell with m sides by its construction, and it will be the m-gon with the fewest possible sides to do so. See A342222 for a proof. This sequence lists the smallest m-sided polygon to contain an n-sided cell where this central cell is not considered for odd m.
See A342222 for other images of the m-sided polygons.
a(17) is presently unknown, but if a(17) > 0 it is greater than 765.

Links

Crossrefs

Cf. A342222, A007678, A331450, A331451, A007569, A135565.
See also A341729 and A341730 for the maximum number of sides in any cell.

Extensions

a(15)-a(16) added by Scott R. Shannon, Mar 15 2021
Minimum value for a(17) updated by Scott R. Shannon, Mar 21 2021
Minimum value for a(17) updated by Scott R. Shannon, Nov 30 2021

A341729 Maximum number of sides in any cell in a regular n-gon with all diagonals drawn (cf. A007678).

Original entry on oeis.org

3, 3, 5, 4, 7, 4, 9, 5, 11, 4, 13, 5, 15, 6, 17, 7, 19, 8, 21, 8, 23, 6, 25, 8, 27, 8, 29, 6, 31, 7, 33, 8, 35, 8, 37, 8, 39, 12, 41, 8, 43, 10, 45, 8, 47, 10, 49, 10, 51, 8, 53, 10, 55, 10, 57, 10, 59, 9, 61, 10, 63, 10, 65, 9, 67, 10, 69, 10, 71, 10, 73, 10, 75, 12, 77, 10, 79, 10
Offset: 3

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 06 2021

Keywords

Comments

For a(2*n) see A341730.
Theorem: a(2*n+1) = 2*n+1. For the proof see A342222.
It would be nice to have a bigger b-file.

Links

Crossrefs

Cf. A007678, A062361, A288177, A331450, A331451, A342222.
Bisections: A144396, A341730.

Extensions

a(141) and beyond from Scott R. Shannon, Nov 30 2021

A351045 Irregular table read by rows: row n gives the number of edges with k facing edges for a regular n-gon with all diagonals drawn, with n>=3 and k>=2.

Original entry on oeis.org

3, 4, 0, 4, 5, 0, 10, 0, 5, 6, 0, 18, 12, 6, 7, 0, 28, 14, 21, 14, 7, 8, 0, 56, 48, 24, 9, 0, 54, 54, 72, 72, 18, 0, 9, 10, 0, 80, 160, 120, 20, 11, 0, 88, 154, 198, 198, 55, 0, 0, 0, 11, 12, 0, 240, 336, 168, 13, 0, 130, 260, 507, 390, 91, 104, 0, 0, 0, 0, 13, 14, 0, 266, 616, 644, 140, 42
Offset: 3

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jan 30 2022

Keywords

Comments

The number of facing edges for a given edge is the number of other edges in the one (for edges on the outside of the n-gon) or two polygons that the edge forms a part of. For example, for an edge shared between two adjoined triangles the number of facing edges is four, as it faces two edges in each of the two triangles it forms a part of.
All edges that are on the outside of the n-gon have two facing edges as any such edge belongs to only one (interior) triangle. Thus T(n,2) = n. For odd n the central created n-gon, see A342222, is surrounded by triangles, thus the edges that form this central n-gon have (n-1)+(3-1) = n+1 facing edges, thus T(n,n+1) >= n.
For all n-gons with even n, or odd n if the central n-gon is ignored, the maximum k for which row(n,k) > 0 is unknown, although it is clearly related to the maximum sided cell for all n-gons; see A349784.

Examples

			A hexagon with all diagonals drawn has six edges (those on the outside of the hexagon) which form one side of a single triangle and thus face two edges, eighteen edges that adjoin two triangles and thus face four edges, twelve edges that adjoin a triangle and a quadrilateral and thus face five edges, and six edges that adjoin two quadrilaterals and thus face six edges. Thus the row for n = 6 is [6, 0, 18, 12, 6]. See the attached image.
The table begins:
3;
4, 0, 4;
5, 0, 10, 0, 5;
6, 0, 18, 12, 6;
7, 0, 28, 14, 21, 14, 7;
8, 0, 56, 48, 24;
9, 0, 54, 54, 72, 72, 18, 0, 9;
10, 0, 80, 160, 120, 20;
11, 0, 88, 154, 198, 198, 55, 0, 0, 0, 11;
12, 0, 240, 336, 168;
13, 0, 130, 260, 507, 390, 91, 104, 0, 0, 0, 0, 13;
14, 0, 266, 616, 644, 140, 42;
15, 0, 180, 600, 945, 630, 435, 0, 15, 0, 0, 0, 0, 0, 15;
16, 0, 448, 1056, 960, 576, 32;
17, 0, 238, 816, 1853, 1224, 425, 272, 34, 0, 0, 0, 0, 0, 0, 0, 17;
18, 0, 900, 1836, 1314, 108, 144;
19, 0, 304, 1520, 2717, 2128, 798, 304, 95, 0, 19, 0, 0, 0, 0, 0, 0, 0, 19;
20, 0, 1000, 2120, 3280, 1600, 100, 240;
21, 0, 378, 2352, 4494, 3276, 1365, 252, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21;
22, 0, 1056, 3828, 5258, 1716, 374, 396, 132;
.
.
See the linked file for the table n = 3..100.

Links

Crossrefs

Cf. A135565, A342222, A349784, A344899, A342236, A342222, A342236.

Formula

Sum of row n = A135565(n).
T(n,2) = n.
T(n,n+1) >= n for odd n.

A341730 Maximum number of sides in any cell in a regular 2n-gon with all diagonals drawn (cf. A007678).

Original entry on oeis.org

3, 4, 4, 5, 4, 5, 6, 7, 8, 8, 6, 8, 8, 6, 7, 8, 8, 8, 12, 8, 10, 8, 10, 10, 8, 10, 10, 10, 9, 10, 10, 9, 10, 10, 10, 10, 12, 10, 10, 10, 10, 10, 10, 10, 14, 10, 10, 10, 10, 12, 12, 10, 11, 10, 12, 10, 12, 12, 14, 14, 12, 12, 12, 12, 14, 12, 12
Offset: 2

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 06 2021

Keywords

Comments

A bisection of A341729.
a(46) = 14, corresponding to a regular 92-gon, suggesting that this sequence may be unbounded (cf. A342222). (The Pfetsch-Ziegler web page discusses a similar question for polygons defined by grid points.) It would be nice to have a b-file.
For which values of n is a(n) odd (and why)?

Links

Crossrefs

Cf. A007678, A062361, A331450, A331451, A341729, A342222.

A349784 Maximum number of sides in any cell in a regular n-gon with all diagonals drawn (cf. A007678), excluding the central n-sided cell for odd values of n.

Original entry on oeis.org

3, 3, 4, 5, 4, 6, 5, 6, 4, 8, 5, 7, 6, 8, 7, 8, 8, 7, 8, 8, 6, 8, 8, 8, 8, 11, 6, 10, 7, 8, 8, 9, 8, 10, 8, 8, 12, 10, 8, 14, 10, 9, 8, 10, 10, 10, 10, 12, 8, 12, 10, 10, 10, 10, 10, 12, 9, 12, 10, 12, 10, 12, 9, 10, 10, 10, 10, 11, 10, 12, 10, 12, 12, 10, 10, 10, 10, 10, 10, 10, 10, 12, 10
Offset: 4

Views

Author

Scott R. Shannon and N. J. A. Sloane, Nov 30 2021

Keywords

Comments

As a regular n-gon with an odd number of sides always creates an n-sided cell at its center when all its diagonals are drawn, see A342222, this n-sided cell is not considered for odd n.
Although the behavior of the sequence is unknown as n -> infinity, the data up to n = 765 implies the sequence is possibly bounded. In the range studied the 14-gon is the predominant maximum-sided cell for n > 300.
No n-gon is currently known that produces a cell with 17 sides or 19 sides and above, other than the corresponding central n-sided cell for odd values of n.
See A342222 and A342236 for images of the n-gons.

Examples

			a(4) = 3 as a regular 4-gon (square) creates four 3-gons (triangles) when all its diagonals are drawn.
a(5) = 3 as a regular 5-gon (pentagon) creates ten 3-gons when all its diagonals are drawn. Also created is a central 5-gon but this cell is not considered.
a(6) = 4 as a regular 6-gon (hexagon) creates eighteen 3-gons and six 4-gons when all its diagonals are drawn.
a(7) = 5 as a regular 7-gon (heptagon) creates thirty-five 3-gons, seven 4-gons and seven 5-gons when all its diagonals are drawn. Also created is a central 7-gon but this cell is not considered.

Links

  • Scott R. Shannon, Table of n, a(n) for n = 4..765
  • Scott R. Shannon, Image showing a close-up of the 18-sided cell in the 708-gon. Zoom in to see the vertices marked as white dots around the 18-gon, shown in violet. The bottom three vertices are extremely close together -- if the image were expanded so that the outer two of these vertices were 1cm away from the inner vertex then the size of the entire 708-gon image would be slightly over 12.9 km in diameter.

Crossrefs

Cf. A341729, A342236, A007678, A331450.

A349549 The smallest k such that a regular k-gon with all diagonals drawn contains cells with a total number of sides of 3 through n, or -1 if no such k exists.

Original entry on oeis.org

4, 6, 7, 9, 15, 17, 35, 41, 71, 102, 202, 211, 843
Offset: 3

Views

Author

Scott R. Shannon, Nov 21 2021

Keywords

Comments

If a(15) > 0 it is greater than 765.
Other than the 15-gon, which contains a central 15-gon by its construction, see A342222, the first k-gon to contain a 15-sided cell is the 399-gon, but it does not contain a 13-gon or a 14-gon. The 647-gon contains cells with total sides 3-12, 14, 15, but it does not contain a 13-gon.
The 561-gon contains cells with total sides 3-14, 16 but it does not contain a 15-gon.
Other than the 17-gon itself no k-gon is currently known that contains a 17-sided cell.
The first k-gon to contain an 18-sided cell is the 231-gon, but it does not contain cells with total sides 13, 15-17. Curiously the other known k-gons to contain 18-sided cells are 245, 469, 628, 708, and like the 231-gon, all of these are also missing the 13, 15-17 sided cells.
See the cross-references for images of the k-gons.

Examples

			The number of cells containing m sides, where 3 <= m <= n, is given below for the currently known values of n. For odd k the list of 0's leading up to the single central k-gon is shown as '...'.
.
  n  | k   | number of m-sided cells, 3 <= m <= n
-----------------------------------------------------
  3  | 4   | 4
  4  | 6   | 18, 6
  5  | 7   | 35, 7, 7, 0, 1
  6  | 9   | 90, 36, 18, 9, 0, 0, 1
  7  | 15  | 585, 600, 150, 105, 15, ..., 1
  8  | 17  | 1054, 901, 357, 136, 17, 34, ..., 1
  9  | 35  | 19705, 20475, 8190, 3640, 560, 315, 35, ..., 1
  10 | 41  | 39278, 37064, 16564, 7298, 1025, 656, 123, 41, ..., 1
  11 | 71  | 361319, 359118, 172246, 65604, 10934, 4118, 568, 71, 71, ..., 1
  12 | 102 | 1587732, 1547238, 699414, 222870, 41616, 9486, 306, 918, 102, 102
  13 | 202 | 24468260, 25271008, 11988296, 3828102, 777700, 171296, 16968, \
                                                              6060, 404, 404, 202
  14 | 211 | 28946246, 30389486, 14708177, 4895411, 1025882, 281896, 14981, \
                                                18568, 633, 422, 211, 211, ..., 1
  15 | 843 | 7465441086, 7927237329, 3927037101, 1250023161, 266472300, \
                 50115507, 5487930, 1534260, 44679, 95259, 843, 3372, 843, ..., 1

Links

  • Scott R. Shannon, Image showing a close-up of the 14-sided cell in the 211-gon. Zoom in to see the vertices marked as white dots around the 14-gon, shown in violet. The bottom three vertices are extremely close together -- if the image were expanded so that the outer two of these vertices were 1 cm away from the inner vertex then the size of the entire 211-gon image would be slightly over 3.5 km in diameter.

Crossrefs

Cf. A342222, A342236, A007678, A331450.

Extensions

a(15) from Scott R. Shannon, May 28 2023

A351129 Number of regions in a regular n-gon with all diagonals drawn whose edges all have the same number of facing edges.

Original entry on oeis.org

1, 0, 1, 0, 1, 8, 1, 0, 1, 132, 66, 56, 46, 144, 171, 576, 305, 620, 652, 616, 852, 1296, 1376, 1482, 1891, 1820, 2379, 4530, 3163, 3328, 3532, 4046, 4656, 4896, 6661, 6460, 7411, 7560, 9595, 11676, 10923, 13552, 10936, 13294, 14806, 17232, 17935, 17200, 20452, 20540, 24964, 27270
Offset: 3

Views

Author

Scott R. Shannon and N. J. A. Sloane, Feb 02 2022

Keywords

Comments

See A351045 for details of an edge's count of facing edges in an n-gon with all diagonals drawn.

Examples

			a(5) = 1. A pentagon with all diagonals drawn contains a central pentagon which is surrounded by five other triangles and therefore all its edges have a facing edge count of 6. See the attached image.
a(8) = 8. An octagon with all diagonals drawn contains eight central triangles all of which are surrounded by three other triangles and therefore all their edges have a facing edge count of 4. See the attached image.
a(15) = 46. A 15-gon with all diagonals drawn contains one central 15-gon which is surrounded by triangles, thirty quadrilaterals which are surrounded by other quadrilaterals, and fifteen triangles which are surrounded by pentagons. This gives a total of forty-six regions whose edges all have the same facing edge count. See the attached image.

Links

Crossrefs

Cf. A351045, A135565, A007678, A342222, A349784, A344899, A342236.

A342269 Take a regular (2*n+1)-gon with all diagonals drawn; a(n) = number of edges in next-to-largest cell.

Original entry on oeis.org

3, 5, 6, 6, 8, 7, 8, 8, 7, 8, 8, 8, 11, 10, 8, 9, 10, 8, 10, 14, 9, 10, 10, 12, 12, 10, 10, 12, 12, 12, 12, 10, 10, 11, 12, 12, 10, 10, 10, 10, 12, 10, 10, 12, 13, 12, 12, 11, 12, 12, 12, 12, 12, 10, 10, 12, 14, 14, 12, 12, 12, 10, 12, 12, 14, 12, 10, 12, 12
Offset: 2

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 07 2021

Keywords

Comments

The largest cell has 2*n+1 sides, this is the runner-up.
It can be read off the rows of A331450.
It would be nice to know how fast this sequence grows. Is it unbounded?

Links

Crossrefs

Cf. A331450, A342236, A341735, A341729, A342222.

A349421 Numbers k such that a regular k-gon, when its vertices are connected by infinite lines, creates polygons outside the k-gon with more sides than any polygon inside the k-gon, excluding the central k-sided polygon for odd values of k.

Original entry on oeis.org

14, 19, 29, 32, 33, 39, 52
Offset: 1

Views

Author

Scott R. Shannon, Nov 17 2021

Keywords

Comments

As a regular k-gon with an odd number of sides always creates a k-sided polygon at the center of the k-gon when its vertices are connected by lines (see A342222), this polygon is excluded when considering the polygons inside the k-gon with the maximum number of sides.
If the next term exists it is greater than 100.

Examples

			14 is a term as when a regular 14-gon's vertices are connected by infinite lines fourteen 6-gons are created outside the vertices while the maximum-sided polygons created inside are 5-gons. See the 14-gon image in A344857.
19 is a term as when a regular 19-gon's vertices are connected by infinite lines nineteen 10-gons are created outside the vertices while the maximum-sided polygons created inside, excluding the central 19-gon, are 8-gons. See the 19-gon image in A344857.
32 is a term as when a regular 32-gon's vertices are connected by infinite lines sixty-four 8-gons are created outside the vertices while the maximum-sided polygons created inside are 7-gons. See the linked image.

Links

Crossrefs

Cf. A344938, A344857, A344311, A007678, A342222.

A350718 Number of regions in a regular n-gon with all diagonals drawn whose edges all have a different number of facing edges.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 44, 0, 130, 84, 180, 128, 374, 180, 418, 440, 714, 704, 1104, 624, 1750, 1976, 2484, 2744, 3190, 2880, 3658, 4416, 5280, 6188, 7000, 7128, 8214, 8892, 10296, 10560, 13120, 14028, 16082, 15928, 22140, 20332, 22466, 26112, 27538, 29200, 36924, 36504, 35934, 40284, 41140
Offset: 3

Views

Author

Scott R. Shannon and N. J. A. Sloane, Feb 03 2022

Keywords

Comments

See A351045 for details of an edge's count of facing edges in an n-gon with all diagonals drawn.
For n = 3 to n = 80 the regions with edges all with a different number of facing edges are all triangles or quadrilaterals. The 81-gon is the first n-gon to contain pentagons with this property. The largest number of edges possible for such regions is unknown.

Examples

			a(11) = 44. The 11-gon contains forty-four triangles whose three edges all have a different number of facing edges. This is the first n-gon to contain such regions. See the attached image.

Links

Crossrefs

Cf. A351045, A351129, A135565, A007678, A342222, A349784.
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