A288180 -id:A288180 - cp's OEIS frontend

A333281 Column 2 of triangle in A288180.

13, 37, 121, 265, 587, 1019, 1797, 2823, 4369, 6257, 9001, 12289, 16775, 21905, 28383, 35901, 45463, 56119, 69351, 84167, 101687, 120869, 143777, 168873, 198191, 229771, 266015, 305379, 350673, 399035, 454243, 513619, 579787, 649899, 727927, 810907, 903581

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 15 2020

nonn

For the graphs defined in A331452 and A288187 only the counts for graphs that are one square wide have formulas for regions, edges, and vertices (see A306302, A331757, A331755). For width 2 there are six such sequences (A331766, A331765, A331763; A333279, A333280, A333281). It would be nice to have a formula for any one of them.

See A333279 for illustrations.

Lars Blomberg, Table of n, a(n) for n = 1..200
Hugo Pfoertner, Illustrations of Chamber Complexes up to 5 X 5.

Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

a(10) and beyond from Lars Blomberg, May 23 2021

A288187 Triangle read by rows: T(n,m) (n >= m >= 1) = number of chambers (or regions) formed by drawing the line segments connecting any two of the (n+1) X (m+1) lattice points in an n X m lattice polygon.

Original entry on oeis.org

4, 16, 56, 46, 176, 520, 104, 388, 1152, 2584, 214, 822, 2502, 5700, 12368, 380, 1452, 4392, 9944, 21504, 37400, 648, 2516, 7644, 17380, 37572, 65810, 115532, 1028, 3952, 12120, 27572, 59784, 105128, 184442, 294040, 1562, 6060, 18476, 42066, 91654, 161352, 282754, 450864, 690816

Offset: 1

Hugo Pfoertner, Jun 06 2017

Chambers are counted regardless of their numbers of vertices.

The n X m lattice polygon mentioned in the definition is an n X m grid of square cells, formed using a grid of n+1 X m+1 points. - N. J. A. Sloane, Feb 07 2019

			The diagonals of the 1 X 1 lattice polygon, i.e. the square, cut it into 4 triangles. Therefore T(1,1)=4.
Triangle begins
4,
16, 56,
46, 176, 520,
104, 388, 1152, 2584,
214, 822, 2502, 5700, 12368,
...

Lars Blomberg, Table of n, a(n) for n = 1..325 (The first 25 rows)
Lars Blomberg, Colored illustration for 3 x 3
Lars Blomberg, Colored illustration for 4 X 4
Lars Blomberg, Colored illustration for 5 X 3
Lars Blomberg, Colored illustration for 5 X 5
Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Huntington Tracy Hall, Counterexamples in Discrete Geometry. Dissertation, Department of Mathematics, University of California Berkeley, Fall 2004.
Serkan Hosten, Diane Maclagan, Bernd Sturmfels, Supernormal Vector Configurations, arXiv:math/0105036 [math.CO], 4 May 2001.
Marc E. Pfetsch, Günter M. Ziegler, Large Chambers in a Lattice Polygon (Notes), March 28, 2001, December 13, 2004.
Marc E. Pfetsch, Günter M. Ziegler, Large Chambers in a Lattice Polygon (Notes), March 28, 2001, December 13, 2004. [Cached copy, with permission]
Hugo Pfoertner, Illustrations of Chamber Complexes up to 5 X 5.

Cf. A288177, A288180, A288181.
The first column is A306302. For column 2 see A333279, A333280, A333281.
If the initial points are arranged around a circle rather than a square we get A006533 and A007678.

T(4,1) added from A306302. - N. J. A. Sloane, Feb 07 2019

T(3,3) corrected and rows for n=4..9 added by Max Alekseyev, Apr 05 2019.

A288177 Maximum number of vertices of any convex polygon formed by drawing all line segments connecting any two lattice points of an n X m convex lattice polygon in the plane written as triangle T(n,m), n >= 1, 1 <= m <= n.

Original entry on oeis.org

3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 4, 5, 5, 6, 6, 4, 5, 5, 6, 6, 6, 4, 5, 6, 6, 6, 7, 7, 4, 5, 7, 6, 7, 7, 7, 7, 4, 5, 6, 6, 7, 7, 8, 8, 8, 4, 5, 6, 6, 7, 7, 8, 8, 8, 7, 4, 5, 6, 6, 7, 7, 8, 8, 8, 8, 8, 4, 5, 7, 6, 7, 7, 8, 7, 8, 8, 8, 8, 4, 5, 8, 6, 7, 7, 8, 7, 8, 8, 8, 8, 8, 4, 5, 8, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 4, 5, 8, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 4, 5, 7, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 4, 5, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 4, 5, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 9

Offset: 1

Hugo Pfoertner, Jun 06 2017

The table is given in the section "Results" of the notes by M. E. Pfetsch and G. M. Ziegler, see link.

An n X m convex lattice polygon presumably means an n X m grid of square cells, formed using a grid of n+1 X m+1 points. - N. J. A. Sloane, Feb 07 2019

			Drawing the diagonals in a lattice square of size 1 X 1 produces 4 triangles, so T(1,1)=3.
Triangle begins:
  3;
  4, 4;
  4, 4, 4;
  4, 4, 5, 5;
  4, 5, 5, 6, 6;
  4, 5, 5, 6, 6, 6;
  4, 5, 6, 6, 6, 7, 7;
  ...

Huntington Tracy Hall, Counterexamples in Discrete Geometry. Dissertation UC, Berkeley, Fall 2004.
Serkan Hosten, Diane Maclagan, and Bernd Sturmfels, Supernormal Vector Configurations, arXiv:math/0105036 [math.CO], 4 May 2001.
Marc E. Pfetsch and Günter M. Ziegler, Large Chambers in a Lattice Polygon (Notes), March 28, 2001, December 13, 2004.
Marc E. Pfetsch and Günter M. Ziegler, Large Chambers in a Lattice Polygon (Notes), March 28, 2001, December 13, 2004. [Cached copy, with permission]
Hugo Pfoertner, Illustrations of Chamber Complexes up to 5 X 5.

Cf. A288178 (diagonal of table), A288179, A288180, A288181, A288187.

A333278 Triangle read by rows: T(n,m) (n >= m >= 1) = number of edges in the graph formed by drawing the line segments connecting any two of the (n+1) X (m+1) lattice points in an n X m grid of squares.

Original entry on oeis.org

8, 28, 92, 80, 296, 872, 178, 652, 1922, 4344, 372, 1408, 4256, 9738, 21284, 654, 2470, 7466, 16978, 36922, 64172, 1124, 4312, 13112, 29874, 64800, 113494, 200028, 1782, 6774, 20812, 47402, 103116, 181484, 319516, 509584, 2724, 10428, 31776, 72398, 158352, 279070, 490396, 782096, 1199428

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 15 2020

T(n,m) = A288180(n,m)+A288187(n,m)-1 (Euler).

For the graphs defined in A331452 and A288187 only the counts for graphs that are one square wide have formulas for regions, edges, and vertices (see A306302, A331757, A331755). For width 2 there are six such sequences (A331766, A331765, A331763; A333279, A333280, A333281). It would be nice to have a formula for any one of them.

			Triangle begins:
8,
28, 92,
80, 296, 872,
178, 652, 1922, 4344,
372, 1408, 4256, 9738, 21284,
654, 2470, 7466, 16978, 36922, 64172,
...

Hugo Pfoertner, Illustration of intersection points up to 6 X 6.

Cf. A288180.
For column 1 see A331757. For column 2 see A333279, A333280, A333281.
Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

A333284 Triangle read by rows: T(m,n) (m >= n >= 1) = number of vertices formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

Original entry on oeis.org

5, 13, 37, 35, 129, 405, 75, 289, 933, 2225, 159, 663, 2155, 5157, 11641, 275, 1163, 3793, 9051, 20341, 35677, 477, 2069, 6771, 16129, 36173, 63987, 114409, 755, 3251, 10727, 25635, 57759, 102845, 183961

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 16 2020

If we only joined pairs of the 2(m+n) boundary points, we would get A331453. If we did not extend the lines to the boundary of the grid, we would get A288180. (One of the links below shows the difference between the three definitions in the case m=3, n=2.)

See A333282 for a large number of colored illustrations.

			Triangle begins:
5,
13, 37,
35, 129, 405,
75, 289, 933, 2225,
159, 663, 2155, 5157, 11641,
275, 1163, 3793, 9051, 20341, 35677,
477, 2069, 6771, 16129, 36173, 63987, 114409,
755, 3251, 10727, 25635, 57759, 102845, 183961, ...
...
T(7,7) corrected Mar 19 2020

Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008 [Local copy]
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009 [Local copy]
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010 [Local copy]
N. J. A. Sloane, Illustration of T(3,2) = 129. [Black lines correspond to A331453(3,2), black + red lines correspond to A288180(3,2), and black + red + blue lines to T(3,2)]
N. J. A. Sloane, Illustration of T(3,3) = 405 [Black lines correspond to A288180(3,3), and black + red lines to T(3,3)]

Cf. A288187, A331452, A288180, A331453, A333282 (regions), A333283 (edges). Column 1 is A331755. The main diagonal is A333285.

More terms and corrections from Scott R. Shannon, Mar 21 2020

A343993 Number of nodes in the graph AC(n,n).

Original entry on oeis.org

5, 37, 353, 1761, 8917, 26773, 84497, 215545, 508613, 1044825, 2184125, 4036481, 7550669, 13068897, 21666789, 34540637, 55656469, 84953609, 130542493, 192747673, 278927225, 393687369, 559130085, 770515489, 1059144047

Offset: 1

N. J. A. Sloane, May 17 2021

nonn

Taken from Lars Blomberg's b-file for A288180.

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.

Main diagonal of triangle A288180.

cp's OEIS Frontend

A333281 Column 2 of triangle in A288180.

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A288187 Triangle read by rows: T(n,m) (n >= m >= 1) = number of chambers (or regions) formed by drawing the line segments connecting any two of the (n+1) X (m+1) lattice points in an n X m lattice polygon.

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A288177 Maximum number of vertices of any convex polygon formed by drawing all line segments connecting any two lattice points of an n X m convex lattice polygon in the plane written as triangle T(n,m), n >= 1, 1 <= m <= n.

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A333278 Triangle read by rows: T(n,m) (n >= m >= 1) = number of edges in the graph formed by drawing the line segments connecting any two of the (n+1) X (m+1) lattice points in an n X m grid of squares.

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A333284 Triangle read by rows: T(m,n) (m >= n >= 1) = number of vertices formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

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Examples

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Crossrefs

Extensions

A343993 Number of nodes in the graph AC(n,n).

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Author

Keywords

Comments

Links

Crossrefs