A338886 a(n) is the number of positive integers k such that there exists a diagonal lattice rectangle touching all four sides of an n X k rectangle.

0, 1, 1, 2, 3, 3, 5, 6, 7, 9, 12, 11, 15, 15, 16, 19, 24, 20, 28, 25, 29, 30, 36, 33, 44, 40, 42, 41, 51, 44, 59, 52, 55, 57, 69, 56, 76, 68, 71, 73, 89, 72, 92, 81, 89, 90, 107, 86, 115, 101, 107, 101, 129, 103, 126, 117, 122, 126, 147, 113, 153, 136, 148

Offset: 1

Author

Peter Kagey, Nov 14 2020

Keywords

• nonn

Comments

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

This sequence gives the row lengths of A338885.

Examples

			For n = 5 there are a(5) = 3 different y-values that appear in the coordinates of diagonal lattice rectangles that touch the x-axis, the y-axis, and the line x = 5. An example of each, listed by vertices counterclockwise:
   y_max = 4: (4,4), (0,2), (1,0), (5,2);
   y_max = 5: (4,5), (0,4), (1,0), (5,1);
   y_max = 7: (3,7), (0,6), (2,0), (5,1).
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..1000
• Code Golf Stack Exchange, Rectangles in rectangles

Crossrefs

Cf. A085582, A113751, A338885, A338887.

Formula

a(n) >= A338887(n).