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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.

%I A356435 #30 Feb 17 2025 08:30:02
%S A356435 0,2,4,8,10,14,16,20,22,26,29,32,32,39,41,44,46,51,52,56,58,62,66,69,
%T A356435 69,74,79,82,85,88,88,92,96,100,103,106,108,113,116,119,120,122,124,
%U A356435 132,135,138,141,143,145,146,152,158,160,164,164,166,172,175,179,181,184,186,189,193,194,199
%N 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.
%C A356435 a(n) <= A057655(n).
%C A356435 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.
%F A356435 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.
%e A356435 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.
%Y A356435 Cf. A057655, A123689, A291259.
%K A356435 nonn
%O A356435 0,2
%A A356435 _Bernard Montaron_, Aug 07 2022