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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.

%I A356462 #22 Feb 17 2025 08:30:13
%S A356462 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 A356462 78,81,89,89,92,97,97,100,104,112,112,115,116,121,122,127,129,137,137,
%U A356462 140,144,148,148,152,155,157,161,164,169,177,177
%N 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.
%C A356462 a(n) >= A057655(n).
%C A356462 The terms of square index of this sequence are such that a(n^2) = A123690(2n), e.g., a(9) = 32 = A123690(6).
%F A356462 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.
%e A356462 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 A356462 Cf. A057655, A123690, A346993.
%K A356462 nonn
%O A356462 0,2
%A A356462 _Bernard Montaron_, Aug 08 2022