A354703 T(w,h) = w*h - A354702(w,h) is a lower bound on the gain in the number of not covered grid points from an optimally positioned and rotated cover versus a just translated cover, where T(w,h) and A354702 are triangles read by rows.

1, 1, 2, 1, 2, 2, 2, 3, 3, 4, 2, 3, 2, 3, 4, 2, 4, 3, 4, 4, 4, 3, 5, 3, 6, 4, 6, 9, 3, 5, 4, 5, 4, 4, 7, 7, 3, 6, 3, 6, 4, 6, 9, 6, 9, 3, 6, 4, 5, 4, 5, 7, 6, 6, 4, 4, 7, 5, 7, 5, 6, 10, 7, 9, 5, 9, 4, 8, 5, 8, 5, 8, 12, 8, 12, 8, 12, 16, 4, 8, 5, 7, 4, 6, 10, 6, 9, 4, 8, 12, 7

Offset: 1

Author

Hugo Pfoertner, Jun 15 2022

Keywords

• nonn
• tabl

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, 2, 2; |  |  |   |  |   |  |   |   |  |
   4 |  2, 3, 3, 4; |  |   |  |   |  |   |   |  |
   5 |  2, 3, 2, 3, 4; |   |  |   |  |   |   |  |
   6 |  2, 4, 3, 4, 4, 4;  |  |   |  |   |   |  |
   7 |  3, 5, 3, 6, 4, 6,  9; |   |  |   |   |  |
   8 |  3, 5, 4, 5, 4, 4,  7, 7;  |  |   |   |  |
   9 |  3, 6, 3, 6, 4, 6,  9, 6,  9; |   |   |  |
  10 |  3, 6, 4, 5, 4, 5,  7, 6,  6, 4;  |   |  |
  11 |  4, 7, 5, 7, 5, 6, 10, 7,  9, 5,  9;  |  |
  12 |  4, 8, 5, 8, 5, 8, 12, 8, 12, 8, 12, 16; |
  13 |  4, 8, 5, 7, 4, 6, 10, 6,  9, 4,  8, 12, 7
.
T(4,3) = 3, because the optimally positioned and rotated 4 X 3 rectangle
covers A354702(4,3) = 9 grid points, whereas a translated, but unrotated 4 X 3 rectangle covers 4*3 = 12 grid points. 4*3 - 9 = 3.
  + . . . . + . . . . + . . . . + . . . . + . . . . + . . . . +
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  .     O---------o---------o---------o---------O   .
  .     |   .         .         .         .     |   .
  + . . | . 1 . . . . 2 . . . . 3 . . . . 4 . . | . +
  .     |   .         .         .         .     |   .
  .     |   .         .         .         .     |   .
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  + . . | . 5 . . . . 6 . . . . 7 . . . . 8 . . | . +
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  .     |   .         .         .         .     |   .
  .     o   .         .         .         .     o   .
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  + . . | . 9 . . . .10 . . . .11 . . . .12 . . | . +
  .     |   .         .         .         .     |   .
  .     |   .         .         .         .     |   .
  .     O---------o---------o---------o---------O   .
  .         .         .         .         .         .
  + . . . . + . . . . + . . . . + . . . . + . . . . +
		

Links

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

Crossrefs

Cf. A354702, A354492 (diagonal).

Cf. A354704, A354705 (similar, but for maximizing the number of covered points).