%I #20 Aug 13 2022 15:45:59

%S 1,5,9,12,14,21,21,24,28,32,37,37,41,45,48,52,52,57,61,63,69,69,72,76,

%T 78,81,89,89,92,97,97,100,104,112,112,115,116,121,122,127,129,137,137,

%U 140,144,148,148,152,155,157,161,164,169,177,177

%N 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.

%C a(n) >= A057655(n).

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

%F 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 symetries 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.

%e 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.

%Y Cf. A057655, A123690, A346993.

%K nonn

%O 0,2

%A _Bernard Montaron_, Aug 08 2022