A356462 - OEIS

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.

1, 5, 9, 12, 14, 21, 21, 24, 28, 32, 37, 37, 41, 45, 48, 52, 52, 57, 61, 63, 69, 69, 72, 76, 78, 81, 89, 89, 92, 97, 97, 100, 104, 112, 112, 115, 116, 121, 122, 127, 129, 137, 137, 140, 144, 148, 148, 152, 155, 157, 161, 164, 169, 177, 177

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OFFSET

0,2

COMMENTS

a(n) >= A057655(n).

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

LINKS

Table of n, a(n) for n=0..54.

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 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.

EXAMPLE

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.

CROSSREFS

Cf. A057655, A123690, A346993.

Sequence in context: A143834 A314612 A314613 * A102183 A153044 A106635

Adjacent sequences: A356459 A356460 A356461 * A356463 A356464 A356465

KEYWORD

nonn

AUTHOR

Bernard Montaron, Aug 08 2022

STATUS

approved