A354707 -id:A354707 - 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.
```

A354705 T(w,h) = (w+1)*(h+1) - A354704(w,h) is an upper bound for the deficit in the number of grid points covered by an optimally positioned and rotated cover compared to the excluded singular case of an axis-parallel, unshifted cover, where T(w,h) and A354704 are triangles read by rows.

Original entry on oeis.org

2, 3, 4, 3, 4, 3, 4, 5, 5, 7, 4, 6, 4, 6, 4, 5, 7, 5, 8, 6, 8, 6, 7, 7, 10, 8, 11, 11, 6, 8, 6, 9, 6, 9, 12, 9, 7, 9, 7, 11, 8, 11, 12, 12, 11, 7, 10, 6, 10, 6, 9, 13, 9, 12, 8, 8, 11, 8, 12, 8, 12, 15, 12, 15, 12, 16, 9, 11, 9, 13, 10, 14, 16, 15, 16, 15, 19, 20

Offset: 1

Hugo Pfoertner, Jun 15 2022

See A354707 for an interpretation of the diagonal terms.

			The triangle begins:
    \ h 1   2  3   4   5   6   7   8   9  10  11  12
   w \ ---------------------------------------------
   1 |  2;  |  |   |   |   |   |   |   |   |   |   |
   2 |  3,  4; |   |   |   |   |   |   |   |   |   |
   3 |  3,  4, 3;  |   |   |   |   |   |   |   |   |
   4 |  4,  5, 5,  7;  |   |   |   |   |   |   |   |
   5 |  4,  6, 4,  6,  4;  |   |   |   |   |   |   |
   6 |  5,  7, 5,  8,  6,  8;  |   |   |   |   |   |
   7 |  6,  7, 7, 10,  8, 11, 11;  |   |   |   |   |
   8 |  6,  8, 6,  9,  6,  9, 12,  9;  |   |   |   |
   9 |  7,  9, 7, 11,  8, 11, 12, 12, 11;  |   |   |
  10 |  7, 10, 6, 10,  6,  9, 13,  9, 12,  8;  |   |
  11 |  8, 11, 8, 12,  8, 12, 15, 12, 15, 12, 16;  |
  12 |  9, 11, 9, 13, 10, 14, 16, 15, 16, 15, 19, 20

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

Cf. A354707 (diagonal).

Cf. A354702, A354703 (similar, but for minimizing the number of covered points), A354704.

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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PARI

A354705 T(w,h) = (w+1)*(h+1) - A354704(w,h) is an upper bound for the deficit in the number of grid points covered by an optimally positioned and rotated cover compared to the excluded singular case of an axis-parallel, unshifted cover, where T(w,h) and A354704 are triangles read by rows.

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A354492 Diagonal of A354703.

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A354706 Diagonal of A354704.

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