A338885 Irregular triangle read by rows in which the n-th row lists all numbers k such that there exists a diagonal lattice rectangle touching all four sides of an n X k rectangle.

2, 3, 4, 5, 4, 5, 7, 6, 9, 10, 5, 7, 8, 11, 13, 7, 8, 10, 13, 16, 17, 6, 9, 11, 12, 15, 19, 21, 6, 8, 10, 11, 14, 17, 22, 25, 26, 7, 9, 10, 11, 13, 14, 16, 17, 19, 25, 29, 31, 9, 12, 13, 15, 18, 20, 21, 28, 33, 36, 37, 7, 8, 11, 12, 13, 14, 15, 17, 20, 22, 23

Offset: 2

Author

Peter Kagey, Nov 14 2020

Keywords

• nonn
• tabf

Comments

A diagonal lattice rectangle is a rectangle with integer coordinates and no side parallel to the x-axis.

Conjecture: The smallest number in the n-th row is A228286(n).

Conjecture: The largest number in the n-th row is A033638(n).

Examples

			Table begins:
   n | n-th row
-----+------------------------------------------------
   2 | 2
   3 | 3
   4 | 4,  5
   5 | 4,  5,  7
   6 | 6,  9, 10
   7 | 5,  7,  8, 11, 13
   8 | 7,  8, 10, 13, 16, 17
   9 | 6,  9, 11, 12, 15, 19, 21
  10 | 6,  8, 10, 11, 14, 17, 22, 25, 26
  11 | 7,  9, 10, 11, 13, 14, 16, 17, 19, 25, 29, 31
  12 | 9, 12, 13, 15, 18, 20, 21, 28, 33, 36, 37
For n = 6, three of the diagonal lattice rectangles that touch the y-axis, x-axis, and line x = 6 are:
(2 ,6), (0,2), (4,0), (6,4);
(2, 9), (0,8), (4,0), (6,1); and
(3,10), (0,9), (3,0), (6,1);
which have maximum y-values of 6, 9, and 10 respectively.
		

Links

• Peter Kagey, Table of n, a(n) for n = 2..11808 (first 100 rows, flattened)
• Code Golf Stack Exchange, Rectangles in rectangles

Crossrefs

Cf. A033638, A085582, A113751, A228286.

Cf. A338886 (row lengths).