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

1,6

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.

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

Example

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

Crossrefs

Cf. A354702.

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

Sequence in context: A361690 A020906 A220280 * A191774 A262885 A097305

Adjacent sequences: A355239 A355240 A355241 * A355243 A355244 A355245

Keyword

tabl,sign

Author

Hugo Pfoertner, Jun 25 2022

Status

approved