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

1,2

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

Table of n, a(n) for n=1..26.

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.

Sequence in context: A317611 A220461 A220287 * A320195 A129163 A083549

Adjacent sequences: A354489 A354490 A354491 * A354493 A354494 A354495

Keyword

nonn,hard,more

Author

Hugo Pfoertner, Jun 22 2022

Status

approved