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

1,3

Links

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

Example

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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  .  \      .         .         .         .       \ .         .
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  + . .\. . 4 . . . . 5 . . . . 6 . . . . 7 . . . . \ . . . . +
  .     o   .         .         .         .         .\        .
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  .       \ .         .         .         .      ~  .         .
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  + . . . . \ . . . . 8 . . . . 9 . . . . + . . . . + . . . . +
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  .         .         .         .         .         .
  .     O---------o---------o---------o---------O   .
  .     |   .         .         .         .     |   .
  + . . | . 1 . . . . 2 . . . . 3 . . . . 4 . . | . +
  .     |   .         .         .         .     |   .
  .     |   .         .         .         .     |   .
  .     o   .         .         .         .     o   .
  .     |   .         .         .         .     |   .
  + . . | . 5 . . . . 6 . . . . 7 . . . . 8 . . | . +
  .     |   .         .         .         .     |   .
  .     |   .         .         .         .     |   .
  .     o   .         .         .         .     o   .
  .     |   .         .         .         .     |   .
  + . . | . 9 . . . .10 . . . .11 . . . .12 . . | . +
  .     |   .         .         .         .     |   .
  .     |   .         .         .         .     |   .
  .     O---------o---------o---------o---------O   .
  .         .         .         .         .         .
  + . . . . + . . . . + . . . . + . . . . + . . . . +

Crossrefs

Cf. A354702, A354492 (diagonal).

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

Sequence in context: A029258 A280128 A367055 * A070096 A319821 A024154

Adjacent sequences: A354700 A354701 A354702 * A354704 A354705 A354706

Keyword

nonn,tabl

Author

Hugo Pfoertner, Jun 15 2022

Status

approved