A356435 a(n) is the minimum 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.

0, 2, 4, 8, 10, 14, 16, 20, 22, 26, 29, 32, 32, 39, 41, 44, 46, 51, 52, 56, 58, 62, 66, 69, 69, 74, 79, 82, 85, 88, 88, 92, 96, 100, 103, 106, 108, 113, 116, 119, 120, 122, 124, 132, 135, 138, 141, 143, 145, 146, 152, 158, 160, 164, 164, 166, 172, 175, 179, 181, 184, 186, 189, 193, 194, 199

Offset: 0

Author

Bernard Montaron, Aug 07 2022

Keywords

• nonn

Comments

a(n) <= A057655(n).

The terms of square index of this sequence are such that a(n^2) = A123689(2n) >= A291259(n), e.g., a(9) = 26 = A123689(6) >= A291259(3) = 25.

Examples

			For n = 1 the minimum number of Z x Z lattice points inside the circle is a(1) = 2. The minimum is obtained, for example, with the circle centered at x = 0.1, y = 0.
		

Crossrefs

Cf. A057655, A123689, A291259.

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 minimum of N(u,v,n) taken over 0 <= u <= 1/2 and 0 <= v <= u. Due to the symmetries of the square lattice one can limit the position (u,v) of the circle center within this triangle. The terms of the sequence were found by "brute force" search of the minimum of N(u,v,n) for (u,v) running through the triangular domain above.