A356462 a(n) is the maximum number of Z x Z lattice points inside or on a circle of radius n^(1/2) for any position of the center of the circle.

1, 5, 9, 12, 14, 21, 21, 24, 28, 32, 37, 37, 41, 45, 48, 52, 52, 57, 61, 63, 69, 69, 72, 76, 78, 81, 89, 89, 92, 97, 97, 100, 104, 112, 112, 115, 116, 121, 122, 127, 129, 137, 137, 140, 144, 148, 148, 152, 155, 157, 161, 164, 169, 177, 177

Offset: 0

Author

Bernard Montaron, Aug 08 2022

Keywords

• nonn

Comments

a(n) >= A057655(n).

The terms of square index of this sequence are such that a(n^2) = A123690(2n), e.g., a(9) = 32 = A123690(6).

Examples

			For n = 1 the maximum number of Z x Z lattice points inside the circle is a(1) = 5. The maximum is obtained with the circle centered at x = 0, y = 0.
		

Crossrefs

Cf. A057655, A123690, A346993.

Formula

Let N(u,v,n) be the number of integer solutions (x,y) of (x-u)^2 + (y-v)^2 <= n. Then a(n) is the maximum of N(u,v,n) taken over 0 <= u <= 1/2 and 0 <= v <= u. The symmetries of the square lattice allow to limit the domain of the circle center (u,v) to this triangle. The terms of this sequence were found by "brute force" search of the maximum of N(u,v,n) for (u,v) in this triangular domain.