A335681 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of interior vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 7, 6, 0, 0, 10, 14, 14, 10, 0, 0, 15, 22, 27, 22, 15, 0, 0, 21, 33, 42, 42, 33, 21, 0, 0, 28, 45, 63, 65, 63, 45, 28, 0, 0, 36, 60, 85, 98, 98, 85, 60, 36, 0, 0, 45, 76, 113, 131, 147, 131, 113, 76, 45, 0, 0, 55, 95, 143, 174, 196, 196, 174, 143, 95, 55, 0

Offset: 1

Author

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020

Keywords

• nonn
• tabl

Comments

The case m=n (the main diagonal) is dealt with in A331755. A306302 has illustrations for the diagonal case for m = 1 to 15.

Also A335678 has colored illustrations for many values of m and n.

Examples

			The initial rows of the array are:
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
  0, 3, 7, 14, 22, 33, 45, 60, 76, 95, 115, 138, ...
  0, 6, 14, 27, 42, 63, 85, 113, 143, 178, 215, 258, ...
  0, 10, 22, 42, 65, 98, 131, 174, 220, 274, 330, 396, ...
  0, 15, 33, 63, 98, 147, 196, 260, 329, 410, 493, 591, ...
  0, 21, 45, 85, 131, 196, 261, 347, 439, 547, 656, 786, ...
  0, 28, 60, 113, 174, 260, 347, 461, 583, 726, 870, 1042, ...
  0, 36, 76, 143, 220, 329, 439, 583, 737, 918, 1099, 1316, ...
  0, 45, 95, 178, 274, 410, 547, 726, 918, 1143, 1368, 1638, ...
  0, 55, 115, 215, 330, 493, 656, 870, 1099, 1368, 1637, 1961, ...
  ...
The initial antidiagonals are:
  0
  0, 0
  0, 1, 0
  0, 3, 3, 0
  0, 6, 7, 6, 0
  0, 10, 14, 14, 10, 0
  0, 15, 22, 27, 22, 15, 0
  0, 21, 33, 42, 42, 33, 21, 0
  0, 28, 45, 63, 65, 63, 45, 28, 0
  0, 36, 60, 85, 98, 98, 85, 60, 36, 0
  0, 45, 76, 113, 131, 147, 131, 113, 76, 45, 0
  0, 55, 95, 143, 174, 196, 196, 174, 143, 95, 55, 0
  ...
		

Links

• Index entries for sequences related to stained glass windows

Crossrefs

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.

For the diagonal case see A306302 and A331755.

Formula

It follows from the definitions that T(m,n) = A335680(m,n) - m - n. Note that there is an explicit formula for the latter sequence.