A354492 Diagonal of A354703.

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

Author

Hugo Pfoertner, Jun 22 2022

Keywords

• nonn
• hard
• more

Comments

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.

Links

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

Crossrefs

Cf. A084068, A293330, A354702, A354703.

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