A341729 -id:A341729 - cp's OEIS frontend

A331450 Irregular triangle read by rows: Take a regular n-sided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., max_k.

1, 4, 10, 0, 1, 18, 6, 35, 7, 7, 0, 1, 56, 24, 90, 36, 18, 9, 0, 0, 1, 120, 90, 10, 176, 132, 44, 22, 276, 168, 377, 234, 117, 39, 0, 13, 0, 0, 0, 0, 1, 476, 378, 98, 585, 600, 150, 105, 15, 0, 0, 0, 0, 0, 0, 0, 1, 848, 672, 128, 48, 1054, 901, 357, 136, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1404, 954, 72, 18, 18, 1653, 1444, 646, 190, 57, 38, 2200, 1580, 580, 120, 0, 20, 2268, 2520, 903, 462, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Jan 25 2020

Computed by Scott R. Shannon, Jan 24 2020
See A331451 for a version of this triangle giving the counts for k = 3 through n.
Mitosis of convex polygons, by Scott R. Shannon and N. J. A. Sloane, Dec 13 2021 (Start)
Borrowing a term from biology, we can think of this process as the "mitosis" of a regular polygon. Row 6 of this triangle shows that a regular hexagon "mitoses" into 18 triangles and 4 quadrilaterals, which we denote by 3^18 4^6.
What if we start with a convex but not necessarily regular n-gon? Let M(n) denote the number of different decompositions into cells that can be obtained. For n = 3, 4, and 5 there is only one possibility. For n = 6 there are two possibilities, 3^18 4^6 and 3^19 4^3 5^3. For n = 7 there are at least 11 possibilities. So the sequence M(n) for n >= 3 begins 1, 1, 1, 2, >=11, ...
The links below give further information. See also A350000. (End)

			A hexagon with all diagonals drawn contains 18 triangles and 6 quadrilaterals, so row 6 is [18, 6].
Triangle begins:
  1,
  4,
  10, 0, 1,
  18, 6,
  35, 7, 7, 0, 1,
  56, 24,
  90, 36, 18, 9, 0, 0, 1,
  120, 90, 10,
  176, 132, 44, 22, 0, 0, 0, 0, 1
  276, 168,
  377, 234, 117, 39, 0, 13, 0, 0, 0, 0, 1,
  476, 378, 98,
  585, 600, 150, 105, 15, 0, 0, 0, 0, 0, 0, 0, 1,
  848, 672, 128, 48,
  1054, 901, 357, 136, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  1404, 954, 72, 18, 18,
  1653, 1444, 646, 190, 57, 38, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
  2200, 1580, 580, 120, 0, 20,
  2268, 2520, 903, 462, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  2992, 2860, 814, 66, 44, 44,
  3749, 2990, 1564, 644, 115, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  ...
The row sums are A007678, the first column is A062361.

M. Rubinstein, Drawings of A007678 for n=4,5,6,....
Scott R. Shannon, Rows 3 through 596 of A331450.
N. J. A. Sloane, Illustration for row n=9 of A331450. [9-gon with one representative for each type of polygonal cell labeled with its number of sides].
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
Scott R. Shannon and N. J. A. Sloane, N. J. A. Sloane, Table showing number of ways to "mitose" a convex n-gon. [From here on the links are related to the mitosis problem, and are in logical rather than alphabetical order]
Scott R. Shannon, Illustration for mitosis 5.1 of a pentagon (the cell counts are the same whether of not the pentagon is regular)
Scott R. Shannon, Illustration for mitosis 6.1 of a hexagon with a triple point (the cell counts are the same whether of not the hexagon is regular, as long as it has a triple point)
Scott R. Shannon, Illustration for mitosis 6.2 of a hexagon (without a triple point)
N. J. A. Sloane, Illustration for mitosis 7a of a 7-gon
Scott R. Shannon, A second (colored) illustration for mitosis 7a of a 7-gon
Scott R. Shannon, Illustration for mitosis 7b
Scott R. Shannon, Illustration for mitosis 7c
Scott R. Shannon, Illustration for mitosis 7d
Scott R. Shannon, Illustration for mitosis 7e
Scott R. Shannon, Illustration for mitosis 7f
Scott R. Shannon, Illustration for mitosis 7g
Scott R. Shannon, Illustration 1 for mitosis 7h
Scott R. Shannon, Illustration 2 for mitosis 7h (same cell counts as preceding illustration but a different polygon - look at the brown cells)
Scott R. Shannon, Illustration for mitosis 7i
Scott R. Shannon, Illustration for mitosis 7j
Scott R. Shannon, Illustration for mitosis 7k
M. F. Hasler, Nine examples of dissections of convex 7-gons [These are all subsumed in the above illustrations]

Cf. A007678, A062361, A331451, A341729, A341730, A335000.

Added "regular" to definition. - N. J. A. Sloane, Mar 06 2021

A342222 a(n) is the smallest m such that a regular m-gon with all diagonals drawn contains a cell with n sides, or a(n) = -1 if no such m exists.

Original entry on oeis.org

3, 6, 5, 9, 7, 13, 9, 29, 11, 40, 13, 43, 15, 212, 17, 231, 19

Offset: 3

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

Theorem: If n is odd then a(n) = n.
Proof. (i) If n is odd then the central cell in a regular n-gon with all diagonals drawn is a smaller regular n-gon. So if n is odd, then a(n) <= n.
(ii) Suppose a convex m-gon, not necessarily regular, with all diagonals drawn has a cell with e edges. Each edge when extended meets two vertices, so at most 2e vertices are involved in defining the boundary of that cell.
On the other hand no vertex can define more than two edges of the cell, so 2e <= 2m, so e <= m. So to get an n-sided cell, we need at least n vertices. So a(n) >= n. QED.
If a(20) > 0 it is greater than 765 - Scott R. Shannon, Nov 30 2021

			Examining the images in A007678, for example Michael Rubinstein's illustration, or the images shown here, we see that the first occurrence of a five-sided cell is for m = 5, so a(5) = 5. The first time we see a four-sided cell is for m = 6, so a(4) = 6.

Martin Balko, Anna Brötzner, Fabian Klute, and Josef Tkadlec, Faces in Rectilinear Drawings of Complete Graphs, 40th European Workshop on Computational Geometry, Ioannina, Greece, March 13-15, 2024. See pp. 4, 7.
Scott R. Shannon, Image for a(3) = 3.
Scott R. Shannon, Image for a(4) = 6.
Scott R. Shannon, Image for a(5) = 5.
Scott R. Shannon, Image for a(6) = a(9) = 9.
Scott R. Shannon, Image for a(7) = 7.
Scott R. Shannon, Image for a(8) = a(13) = 13.
Scott R. Shannon, Image for a(10) = 29.
Scott R. Shannon, Image for a(11) = 11.
Scott R. Shannon, Image for a(12) = 40.
Scott R. Shannon, Image for a(14) = 43.
Scott R. Shannon, Image for a(15) = 15.

Cf. A007678, A331450, A331451, A007569, A135565.

See also A341729 and A341730 for the maximum number of sides in any cell.

a(16)-a(19) added by Scott R. Shannon, Mar 14 2021

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

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

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.

Scott R. Shannon, Image for a(3) = 4.
Scott R. Shannon, Image for a(9) = 35.
Scott R. Shannon, Image for a(13) = 93.

Cf. A342222, A007678, A331450, A331451, A007569, A135565.

See also A341729 and A341730 for the maximum number of sides in any cell.

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

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

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

nonn

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)?

M. E. Pfetsch and G. M. Ziegler, Large chambers in a lattice polygon, December 13, 2004;
M. E. Pfetsch and G. M. Ziegler, Large chambers in a lattice polygon [Local copy of webpage, pdf only]

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

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

nonn

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.

			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.

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.

Cf. A341729, A342236, A007678, A331450.

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

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

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?

Scott R. Shannon, Image for a(2) = 3. The 5-gon has a next-to-largest cell with 3 edges.
Scott R. Shannon, Image for a(3) = 5. The 7-gon has a next-to-largest cell with 5 edges.
Scott R. Shannon, Image for a(4) = 6. The 9-gon has a next-to-largest cell with 6 edges.
Scott R. Shannon, Image for a(7) = 7. The 15-gon has a next-to-largest cell with 7 edges.
Scott R. Shannon, Image for a(6) = 8. The 13-gon has a next-to-largest cell with 8 edges.
Scott R. Shannon, Image for a(17) = 9. The 35-gon has a next-to-largest cell with 9 edges.
Scott R. Shannon, Image for a(15) = 10. The 31-gon has a next-to-largest cell with 10 edges.
Scott R. Shannon, Image for a(14) = 11. The 29-gon has a next-to-largest cell with 11 edges.
Scott R. Shannon, Image for a(25) = 12. The 51-gon has a next-to-largest cell with 12 edges.

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

cp's OEIS Frontend

A331450 Irregular triangle read by rows: Take a regular n-sided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., max_k.

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A342222 a(n) is the smallest m such that a regular m-gon with all diagonals drawn contains a cell with n sides, or a(n) = -1 if no such m exists.

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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.

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

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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.

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

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