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%I #8 Feb 05 2023 02:46:47
%S 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
%N Diagonal of A354703.
%C 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.
%C 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.
%H Hugo Pfoertner, <a href="/A293330/a293330_2.pdf">Illustrations of diagonal terms A354702(1,1)..A354702(25,25)</a>.
%H Hugo Pfoertner, <a href="/A354492/a354492.pdf">Illustration of a(26)</a>.
%Y Cf. A084068, A293330, A354702, A354703.
%Y A354707 is the analogous sequence, but for the problem of maximizing the number of grid points covered.
%K nonn,hard,more
%O 1,2
%A _Hugo Pfoertner_, Jun 22 2022
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