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

1,1

Comments

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.)

Links

Table of n, a(n) for n=1..45.

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)]

Example

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

Crossrefs

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

Sequence in context: A331457 A331452 A288187 * A212520 A115108 A127393

Adjacent sequences: A333279 A333280 A333281 * A333283 A333284 A333285

Keyword

nonn,tabl

Author

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

Extensions

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

More terms from Scott R. Shannon, May 27 2021

Status

approved