A333283 -id:A333283 - 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

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

A333460 Number of edges in an diagonal-edged (or diamond-shaped) checkerboard with width and height 2*n-1 (see Comments in A333434 for definition).

Original entry on oeis.org

8, 172, 1864, 9396, 28188, 72688, 153240, 269680, 500344, 795344, 1158752, 1820428, 2506948, 3477052, 4772708, 6313892, 8116916, 10200156, 13320308, 16199876, 20189488, 24692976, 28950448, 36096712, 41795388, 49216724, 58997920

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 22 2020

See A333434 for the definition and images of the checkerboard.

Cf. A333434 (regions), A333458 (n-gons), A333459 (vertices), A333027, A333036, A333283.

a(8)-a(27) from Lars Blomberg, Jun 03 2020

A333285 The main diagonal of the triangular array A333284.

Original entry on oeis.org

5, 37, 405, 2225, 11641, 35677, 114409, 295701, 718469, 1475709, 3093025, 5771929, 10895273, 18785841, 31414269, 50274501, 81288641, 124066161, 190860537, 282399889, 411505049, 580614301, 824814797, 1138709849, 1570665877, 2115178249, 2833746309, 3732420861, 4937226173

Offset: 1

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

nonn

See A333282, A333283, and A333284 for further information, illustrations, etc.

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, September 02, 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]

Cf. A333282, A333283, A333284.

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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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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Author

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Examples

Links

Crossrefs

Extensions

A333460 Number of edges in an diagonal-edged (or diamond-shaped) checkerboard with width and height 2*n-1 (see Comments in A333434 for definition).

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Extensions

A333285 The main diagonal of the triangular array A333284.

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