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

0,2

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.

Links

Table of n, a(n) for n=0..65.

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 symetries 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.

Example

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.

Sequence in context: A189782 A047235 A328588 * A287844 A219696 A087505

Adjacent sequences: A356432 A356433 A356434 * A356436 A356437 A356438

Keyword

nonn

Author

Bernard Montaron, Aug 07 2022

Status

approved