A293330 -id:A293330 - cp's OEIS frontend

A354704 T(w,h) is a lower bound for the maximum number of grid points in a square grid covered by an arbitrarily positioned and rotated rectangle of width w and height h, excluding the trivial case of an axis-parallel unshifted cover, where T(w,h) is a triangle read by rows.

2, 3, 5, 5, 8, 13, 6, 10, 15, 18, 8, 12, 20, 24, 32, 9, 14, 23, 27, 36, 41, 10, 17, 25, 30, 40, 45, 53, 12, 19, 30, 36, 48, 54, 60, 72, 13, 21, 33, 39, 52, 59, 68, 78, 89, 15, 23, 38, 45, 60, 68, 75, 90, 98, 113, 16, 25, 40, 48, 64, 72, 81, 96, 105, 120, 128, 17, 28, 43, 52, 68, 77, 88, 102, 114, 128, 137, 149

Offset: 1

Hugo Pfoertner, Jun 15 2022

Grid points must lie strictly within the covering rectangle, i.e., grid points on the perimeter of the rectangle are not allowed. See A354702 for more information.

			The triangle begins:
    \ h  1   2   3   4   5   6   7    8    9   10   11   12
   w \ ----------------------------------------------------
   1 |   2;  |   |   |   |   |   |    |    |    |    |    |
   2 |   3,  5;  |   |   |   |   |    |    |    |    |    |
   3 |   5,  8, 13;  |   |   |   |    |    |    |    |    |
   4 |   6, 10, 15, 18;  |   |   |    |    |    |    |    |
   5 |   8, 12, 20, 24, 32;  |   |    |    |    |    |    |
   6 |   9, 14, 23, 27, 36, 41;  |    |    |    |    |    |
   7 |  10, 17, 25, 30, 40, 45, 53;   |    |    |    |    |
   8 |  12, 19, 30, 36, 48, 54, 60,  72;   |    |    |    |
   9 |  13, 21, 33, 39, 52, 59, 68,  78,  89;   |    |    |
  10 |  15, 23, 38, 45, 60, 68, 75,  90,  98, 113;   |    |
  11 |  16, 25, 40, 48, 64, 72, 81,  96, 105, 120, 128;   |
  12 |  17, 28, 43, 52, 68, 77, 88, 102, 114, 128, 137, 149

Hugo Pfoertner, Table of n, a(n) for n = 1..210, rows 1..20 of triangle, flattened
Hugo Pfoertner, Illustrations of the initial terms up to T(5,5).

Cf. A293330, A354702, A354705, A354706 (diagonal), A354707, A355244.

Cf. A123690 (similar problem with circular disks).

PARI
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\\ See links in A354702 and A355244.
```

A354702 T(w,h) is an upper bound for the minimum number of grid points in a square grid covered by an arbitrarily positioned and rotated rectangle of width w and height h, where T(w,h) is a triangle read by rows.

Original entry on oeis.org

0, 1, 2, 2, 4, 7, 2, 5, 9, 12, 3, 7, 13, 17, 21, 4, 8, 15, 20, 26, 32, 4, 9, 18, 22, 31, 36, 40, 5, 11, 20, 27, 36, 44, 49, 57, 6, 12, 24, 30, 41, 48, 54, 66, 72, 7, 14, 26, 35, 46, 55, 63, 74, 84, 96, 7, 15, 28, 37, 50, 60, 67, 81, 90, 105, 112, 8, 16, 31, 40, 55, 64, 72, 88, 96, 112, 120, 128

Offset: 1

Hugo Pfoertner, Jun 15 2022

Grid points must lie strictly within the covering rectangle, i.e., grid points on the perimeter of the rectangle are not allowed.
These upper bounds were determined by an extensive random search, the results of which were stable. The proof that none of these bounds can be improved should be possible with a constructive technique such as integer linear programming applied to all combinatorially possible positions of the rectangle relative to the lattice.
A simple random search is implemented in the attached PARI program, which enables a plausibility check of the results for small covering rectangles. It also provides results for the maximum problem. Additional methods were used to obtain the results shown. In particular, angular orientations of the rectangle along connecting lines between all pairs of lattice points and extreme positions of the rectangle, where lattice points are very close to the corners of the rectangle, were investigated, using adjacent terms in A000404.

			The triangle begins:
    \ h 1   2   3   4   5   6   7   8   9   10   11   12
   w \ -------------------------------------------------
   1 |  0;  |   |   |   |   |   |   |   |    |    |    |
   2 |  1,  2;  |   |   |   |   |   |   |    |    |    |
   3 |  2,  4,  7;  |   |   |   |   |   |    |    |    |
   4 |  2,  5,  9, 12;  |   |   |   |   |    |    |    |
   5 |  3,  7, 13, 17, 21;  |   |   |   |    |    |    |
   6 |  4,  8, 15, 20, 26, 32;  |   |   |    |    |    |
   7 |  4,  9, 18, 22, 31, 36, 40;  |   |    |    |    |
   8 |  5, 11, 20, 27, 36, 44, 49, 57;  |    |    |    |
   9 |  6, 12, 24, 30, 41, 48, 54, 66, 72;   |    |    |
  10 |  7, 14, 26, 35, 46, 55, 63, 74, 84,  96;   |    |
  11 |  7, 15, 28, 37, 50, 60, 67, 81, 90, 105, 112;   |
  12 |  8, 16, 31, 40, 55, 64, 72, 88, 96, 112, 120, 128

Hugo Pfoertner, Table of n, a(n) for n = 1..210, rows 1..20 of triangle, flattened
Hugo Pfoertner, Illustrations of the initial terms up to T(5,5).
Hugo Pfoertner, PARI program

Cf. A293330 (diagonal).
Cf. A354492, A354703, A354704, A354705, A355241, A355242.
Cf. A291259 (similar problem for circular disks).
Cf. A000404 (used to check extreme positions of grid points).

PARI
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\\ See link.
```
PARI
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\\ See also program link in A355241.
```

A354492 Diagonal of A354703.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 9, 7, 9, 4, 9, 16, 7, 16, 8, 14, 9, 12, 23, 13, 21, 8, 17, 32, 20, 28

Offset: 1

Hugo Pfoertner, Jun 22 2022

a(n)-n is an indicator of whether the free space between the covered grid points and the perimeter of the square is relatively large. a(n)-n > 0 for n = 7, 12, 14, 19, 24, 26, ... . A comparison with the linked illustrations from A354702 shows that in all these cases the covering square is rotated by Pi/4 and that the next outer diagonal rows of grid points are very close to the perimeter of the covering square.

In these cases it is favorable if the difference from n*sqrt(2) to the next larger integer is as small as possible. This also fits with 7 and 12 being terms in A084068. Since A084068(5) = 41, it is expected that a record of a(n)-n will occur at a(41) = 41^2 - A354702(41,41) = 1681 - 1624 = 57 and a(n)-n = 16.

Hugo Pfoertner, Illustrations of diagonal terms A354702(1,1)..A354702(25,25).
Hugo Pfoertner, Illustration of a(26).

Cf. A084068, A293330, A354702, A354703.

A354707 is the analogous sequence, but for the problem of maximizing the number of grid points covered.

A354706 Diagonal of A354704.

Original entry on oeis.org

2, 5, 13, 18, 32, 41, 53, 72, 89, 113, 128, 149, 181, 205, 242, 265, 313, 338, 373, 421, 450, 512, 545, 584, 648, 697

Offset: 1

Hugo Pfoertner, Jun 19 2022

a(n) is a lower bound for the maximum number of grid points in a square grid covered by an arbitrarily positioned and rotated square of side length n, excluding the trivial case of an axis-parallel unshifted square.

Grid points must be strictly inside the covering square, i.e., grid points on the perimeter of the square are not allowed.

			For examples see the figures in the linked file.

Hugo Pfoertner, Illustrations of the terms a(1)..a(22).

Cf. A293330, A354704, A354705, A354707.

a(n) = A354704(n,n).

cp's OEIS Frontend

A354704 T(w,h) is a lower bound for the maximum number of grid points in a square grid covered by an arbitrarily positioned and rotated rectangle of width w and height h, excluding the trivial case of an axis-parallel unshifted cover, where T(w,h) is a triangle read by rows.

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A354702 T(w,h) is an upper bound for the minimum number of grid points in a square grid covered by an arbitrarily positioned and rotated rectangle of width w and height h, where T(w,h) is a triangle read by rows.

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PARI

A354492 Diagonal of A354703.

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A354706 Diagonal of A354704.

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