A354702 T(w,h) is an upper bound for the minimum number of grid points in a square grid covered by an arbitrarily positioned and rotated rectangle of width w and height h, where T(w,h) is a triangle read by rows.

0, 1, 2, 2, 4, 7, 2, 5, 9, 12, 3, 7, 13, 17, 21, 4, 8, 15, 20, 26, 32, 4, 9, 18, 22, 31, 36, 40, 5, 11, 20, 27, 36, 44, 49, 57, 6, 12, 24, 30, 41, 48, 54, 66, 72, 7, 14, 26, 35, 46, 55, 63, 74, 84, 96, 7, 15, 28, 37, 50, 60, 67, 81, 90, 105, 112, 8, 16, 31, 40, 55, 64, 72, 88, 96, 112, 120, 128

Offset: 1

Author

Hugo Pfoertner, Jun 15 2022

Keywords

• nonn
• tabl

Comments

Grid points must lie strictly within the covering rectangle, i.e., grid points on the perimeter of the rectangle are not allowed.

These upper bounds were determined by an extensive random search, the results of which were stable. The proof that none of these bounds can be improved should be possible with a constructive technique such as integer linear programming applied to all combinatorially possible positions of the rectangle relative to the lattice.

A simple random search is implemented in the attached PARI program, which enables a plausibility check of the results for small covering rectangles. It also provides results for the maximum problem. Additional methods were used to obtain the results shown. In particular, angular orientations of the rectangle along connecting lines between all pairs of lattice points and extreme positions of the rectangle, where lattice points are very close to the corners of the rectangle, were investigated, using adjacent terms in A000404.

Examples

			The triangle begins:
    \ h 1   2   3   4   5   6   7   8   9   10   11   12
   w \ -------------------------------------------------
   1 |  0;  |   |   |   |   |   |   |   |    |    |    |
   2 |  1,  2;  |   |   |   |   |   |   |    |    |    |
   3 |  2,  4,  7;  |   |   |   |   |   |    |    |    |
   4 |  2,  5,  9, 12;  |   |   |   |   |    |    |    |
   5 |  3,  7, 13, 17, 21;  |   |   |   |    |    |    |
   6 |  4,  8, 15, 20, 26, 32;  |   |   |    |    |    |
   7 |  4,  9, 18, 22, 31, 36, 40;  |   |    |    |    |
   8 |  5, 11, 20, 27, 36, 44, 49, 57;  |    |    |    |
   9 |  6, 12, 24, 30, 41, 48, 54, 66, 72;   |    |    |
  10 |  7, 14, 26, 35, 46, 55, 63, 74, 84,  96;   |    |
  11 |  7, 15, 28, 37, 50, 60, 67, 81, 90, 105, 112;   |
  12 |  8, 16, 31, 40, 55, 64, 72, 88, 96, 112, 120, 128
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..210, rows 1..20 of triangle, flattened
• Hugo Pfoertner, Illustrations of the initial terms up to T(5,5).
• Hugo Pfoertner, PARI program

Crossrefs

Cf. A293330 (diagonal).

Cf. A354492, A354703, A354704, A354705, A355241, A355242.

Cf. A291259 (similar problem for circular disks).

Cf. A000404 (used to check extreme positions of grid points).

Programs

PARI

\\ See link.

PARI

\\ See also program link in A355241.