A355241 T(w,h)/2 is the minimum slope >= 1/2 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 slope satisfying this condition exists.

1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 1, 6, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 6, 1, 2, 1, 2, 2, 1, 2, 6, 2, 2, 2, 2, 2, 2, 1, 1, 6, 6, 2, 1, 2, 1, 2, 2, 1, 2, 6, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 6, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 6, 2, 2, 1, 2, 2, 2, 2, 2

Offset: 1

Author

Hugo Pfoertner, Jun 27 2022

Keywords

• nonn
• tabl

Comments

No example of T(w,h) = -1 is known for w <= 20, i.e., the upper bound A354702(w,h) can always be achieved using a slope that is an integer multiple of 1/2. In the range w <= 20, T(17,13) = 3 is the only occurrence of the required slope 3/2.

For some rectangle dimensions it is possible to reach the value of A354702(w,h) with different slopes. In the simplest case, e.g., with the slopes 1/2 (T(w,h)=1) and 1 (A355242(w,h)=1). The linked file shows examples for some pairs of values (w,h) and the case of (10,10) with 3 different slopes.

Examples

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

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..210, rows 1..20 of triangle, flattened
• Hugo Pfoertner, Different slopes with the same number of grid points covered.
• Hugo Pfoertner, PARI program

Crossrefs

Cf. A354702, A355242.

A355244 is similar, but for maximizing the number of covered grid points.

Programs

PARI

/* See Pfoertner link. The program can be used to validate the given terms by calling it successively with the slope parameter k, starting with k = 1/2, 2/2=1, 3/2, (4/2 = 2 already covered by 1/2 via symmetry), 5/2, 6/2=3 for the desired rectangle size w X h , until the number of grid points given by A354702(w,k) is reached for the first time as a result. Without specifying the slope parameter, the program tries to approximate A354702(w,k) and determine a position of the rectangle maximizing the free space between peripheral grid points and the rectangle. */