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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.
0
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.
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
KEYWORD
nonn
AUTHOR
Bernard Montaron, Aug 07 2022
STATUS
approved