A355242 T(w,h) is the minimum integer slope >= 1 that can be chosen as orientation of a w X h rectangle such that the upper bound for the minimum number of covered grid points A354702(w,d) can be achieved by a suitable translation of the rectangle, where T(w,h) and A354702 are triangles read by rows. T(w,h) = -1 if no integer slope satisfying this condition exists.

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 3, 3, 1, 2, 1, 2, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Author

Hugo Pfoertner, Jun 25 2022

Keywords

• tabl
• sign

Comments

T(17,13) = -1 is the first occurrence of the situation that it is not possible to reach the upper limit A354702(17,13) = 215 with a rectangle whose long side has an integer slope. (17 X 13)-rectangles with integer slope cannot cover less than 216 grid points. To achieve 215 grid points requires a slope of 3/2, i.e. A355241(17,13) = 3. See the linked file for related illustrations.

Examples

			The triangle begins:
    \ h 1  2  3  4  5  6  7  8  9 10 11 12 13
   w \ --------------------------------------
   1 |  1; |  |  |  |  |  |  |  |  |  |  |  |
   2 |  1, 1; |  |  |  |  |  |  |  |  |  |  |
   3 |  1, 1, 2; |  |  |  |  |  |  |  |  |  |
   4 |  1, 1, 2, 1; |  |  |  |  |  |  |  |  |
   5 |  1, 1, 2, 1, 3; |  |  |  |  |  |  |  |
   6 |  1, 1, 2, 1, 3, 1; |  |  |  |  |  |  |
   7 |  1, 1, 1, 1, 1, 1, 1; |  |  |  |  |  |
   8 |  1, 1, 2, 1, 3, 1, 1, 2; |  |  |  |  |
   9 |  1, 1, 1, 1, 3, 1, 1, 1, 1; |  |  |  |
  10 |  1, 1, 2, 1, 3, 3, 1, 2, 1, 2; |  |  |
  11 |  1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1; |  |
  12 |  1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1; |
  13 |  1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..210, rows 1..20 of triangle, flattened
• Hugo Pfoertner, (17 X 13)-rectangles with minimum number of covered grid points

Crossrefs

Cf. A354702.

A355241 is similar, but with slopes chosen from the list 1/2, 1, 3/2, 2, ... .