A288187 -id:A288187 - cp's OEIS frontend

A333282 Triangle read by rows: T(m,n) (m >= n >= 1) = number of regions 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.

4, 16, 56, 46, 192, 624, 104, 428, 1416, 3288, 214, 942, 3178, 7520, 16912, 380, 1672, 5612, 13188, 29588, 51864, 648, 2940, 9926, 23368, 52368, 92518, 164692, 1028, 4624, 15732, 37184, 83628, 148292, 263910, 422792, 1562, 7160, 24310, 57590, 130034, 230856, 410402, 658080, 1023416

Offset: 1

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

Triangle gives number of nodes in graph LC(m,n) in the notation of Blomberg-Shannon-Sloane (2020).

If we only joined pairs of the 2(m+n) boundary points, we would get A331452. If we did not extend the lines to the boundary of the grid, we would get A288187. (One of the links below shows the difference between the three definitions in the case m=3, n=2.)

			Triangle begins:
4,
16, 56,
46, 192, 624,
104, 428, 1416, 3288,
214, 942, 3178, 7520, 16912,
380, 1672, 5612, 13188, 29588, 51864,
648, 2940, 9926, 23368, 52368, 92518, 164692,
1028, 4624, 15732, 37184, 83628, 148292, 263910, 422792
1562, 7160, 24310, 57590, 130034, 230856, 410402, 658080, 1023416
2256, 10336, 35132, 83116, 187376, 331484, 588618, 942808, 1466056, 2101272

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). 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]
Scott R. Shannon, Colored illustration for T(3,2)
Scott R. Shannon, Colored illustration for T(3,2) (edge number coloring)
Scott R. Shannon, Colored illustration for T(3,3)
Scott R. Shannon, Colored illustration for T(3,3) (edge number coloring)
Scott R. Shannon, Colored illustration for T(4,4)
Scott R. Shannon, Colored illustration for T(4,4) (edge number coloring)
Scott R. Shannon, Colored illustration for T(5,5)
Scott R. Shannon, Colored illustration for T(5,5) (edge number coloring)
Scott R. Shannon, Colored illustration for T(6,3)
Scott R. Shannon, Colored illustration for T(6,3) (edge number coloring)
Scott R. Shannon, Colored illustration for T(7,4)
Scott R. Shannon, Colored illustration for T(7,4) (edge number coloring)
Scott R. Shannon, Colored illustration for T(8,2)
Scott R. Shannon, Colored illustration for T(8,2) (edge number coloring)
Scott R. Shannon, Numerical properties of these structures. Corrected version Mar 24 2020.
N. J. A. Sloane, Illustration of T(3,2) = 192. [Black lines correspond to A331452(3,2), black + red lines correspond to A288187(3,2), and black + red + blue lines to T(3,2)]
N. J. A. Sloane, Illustration of T(3,3) = 624 [Black lines correspond to A288187(3,3), and black + red lines to T(3,3)]

Cf. A288187, A331452, A333283 (edges), A333284 (vertices). Column 1 is A306302. Main diagonal is A333294.

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

More terms from Scott R. Shannon, May 27 2021

A333283 Triangle read by rows: T(m,n) (m >= n >= 1) = number of edges 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

8, 28, 92, 80, 320, 1028, 178, 716, 2348, 5512, 372, 1604, 5332, 12676, 28552, 654, 2834, 9404, 22238, 49928, 87540, 1124, 5008, 16696, 39496, 88540, 156504, 279100, 1782, 7874, 26458, 62818, 141386, 251136, 447870

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 A331454. If we did not extend the lines to the boundary of the grid, we would get A333278. (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:
8,
28, 92,
80, 320, 1028,
178, 716, 2348, 5512,
372, 1604, 5332, 12676, 28552,
654, 2834, 9404, 22238, 49928, 87540,
1124, 5008, 16696, 39496, 88540, 156504, 279100,
1782, 7874, 26458, 62818, 141386, 251136, 447870, ...
...
T(7,7) corrected Mar 19 2020

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) = 320. [Black lines correspond to A331454(3,2), black + red lines correspond to A333278(3,2), and black + red + blue lines to T(3,2)]
N. J. A. Sloane, Illustration of T(3,3) = 1028 [Black lines correspond to A288187(3,3), and black + red lines to T(3,3)]

Cf. A288187, A331452, A333278, A331454, A333282 (regions), A333284 (vertices). Column 1 is A331757.

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

A288181 Occurrence counts of chambers with maximum number of vertices in the chamber complex of an n X m lattice polygon as described in A288177, written as triangle T(n,m), n >= 1, 1 <= m <= n.

Original entry on oeis.org

4, 2, 8, 14, 54, 168

Offset: 1

Hugo Pfoertner, Jun 06 2017

			The chamber complex of the 5 X 5 lattice polygon has 16 chambers of size 6, so T(5,5)=16, taken from illustration on Pfetsch and Ziegler web page, see link in A288177.

Hugo Pfoertner, Chamber complex of 5 X 5 lattice polygon. Illustration.
Hugo Pfoertner, Chamber complex of 7 X 5 lattice polygon. Illustration.
Hugo Pfoertner, Chamber complex of 9 X 5 lattice polygon. Illustration.

Cf. A288177, 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

cp's OEIS Frontend

A333282 Triangle read by rows: T(m,n) (m >= n >= 1) = number of regions 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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A333283 Triangle read by rows: T(m,n) (m >= n >= 1) = number of edges 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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A288181 Occurrence counts of chambers with maximum number of vertices in the chamber complex of an n X m lattice polygon as described in A288177, 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions