A355244 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 lower bound for the maximum number of covered grid points A354704(w,d) can be achieved by a suitable translation of the rectangle, where T(w,h) and A354704 are triangles read by rows. T(w,h) = -1 if no slope satisfying this condition exists.

1, 1, 1, 1, 2, 2, 1, 3, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 6, 2, 2, 2, 1, 6, 2, 6, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 6, 2, 1, 2, 2, 1, 3, 2, -1, 2, 2, 3, 2, 1, 2, -1, 3, 2, 1, 2, 2, 2, 2, 6, 2, 1, 2, 2, 1, 2

Offset

1,5

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..210, rows 1..20 of triangle, flattened

Hugo Pfoertner, Illustrations of T(4,2) = 3, T(7,6) = T(9,6) = T(13,12) = 1.

Hugo Pfoertner, Illustrations of T(12,4) = T(12,11) = -1.

Hugo Pfoertner, PARI program

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, 2, 2; |  |  |  |  |  |  |  |  |  |
   4 |  1, 3, 2, 2; |  |  |  |  |  |  |  |  |
   5 |  1, 1, 2, 2, 2; |  |  |  |  |  |  |  |
   6 |  1, 1, 2, 2, 2, 2; |  |  |  |  |  |  |
   7 |  1, 6, 2, 2, 2, 1, 6; |  |  |  |  |  |
   8 |  2, 6, 2, 2, 2, 2, 2, 2; |  |  |  |  |
   9 |  1, 1, 2, 2, 2, 1, 1, 2, 1; |  |  |  |
  10 |  2, 1, 2, 2, 2, 2, 2, 2, 2, 2; |  |  |
  11 |  2, 1, 2, 2, 2, 2, 6, 2, 1, 2, 2; |  |
  12 |  1, 3, 2,-1, 2, 2, 3, 2, 1, 2,-1, 3; |
  13 |  2, 1, 2, 2, 2, 2, 6, 2, 1, 2, 2, 1, 2
.
The first linked illustration shows examples where 2 slopes lead to the same number of covered grid points, where then the smallest multiple of 1/2 is used as a term in the sequence.
The second illustration shows the two examples where it is not possible to cover the maximum number of grid points with a rectangle whose side slope is an integer multiple of 1/2.

Prog

(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 A354704(w, k) is reached for the first time as a result. If the slope parameter is not specified, the program attempts to approximate A354704(w, k) and determine a location of the rectangle that maximizes the free margin between the peripheral grid points and the perimeter of the rectangle. */

Crossrefs

Cf. A354704, A354706.

Cf. A355241 (similar, but with number of covered grid points minimized).

Sequence in context: A124018 A309288 A235061 * A364730 A283004 A111709

Adjacent sequences: A355241 A355242 A355243 * A355245 A355246 A355247

Keyword

tabl,sign,changed

Author

Hugo Pfoertner, Jun 29 2022

Status

approved