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.

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

Author

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

Keywords

• nonn
• tabl
• more

Comments

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.

Examples

			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
		

Links

• 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)]

Crossrefs

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

Extensions

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