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.

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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..35.

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)]

Example

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

Crossrefs

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

Sequence in context: A332599 A331453 A288180 * A141408 A238460 A342475

Adjacent sequences: A333281 A333282 A333283 * A333285 A333286 A333287

Keyword

nonn,tabl,more

Author

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

Extensions

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

Status

approved