A303644 a(n) is the number of lattice points in a Cartesian grid between a square of side length 2*n, centered at the origin, and its inscribed circle. The sides of the square are parallel to the coordinate axes.

0, 0, 0, 4, 4, 12, 24, 32, 40, 48, 68, 92, 100, 120, 136, 168, 192, 220, 244, 268, 312, 336, 376, 420, 444, 484, 524, 576, 624, 664, 724, 764, 820, 868, 912, 992, 1040, 1116, 1156, 1220, 1304, 1368, 1440, 1496, 1564, 1660, 1732, 1816, 1888, 1960, 2032, 2116, 2220, 2308

Offset: 1

Author

Kirill Ustyantsev, Apr 27 2018

Keywords

• nonn

Comments

The borders of the square and the circle are not included. Rotating the square by 45 degrees (so that its vertices lie on the coordinate axes) results in sequence A303646 instead.

Examples

			For n = 4, we have 4 points with integer coordinates; the point in the first quadrant is at (3,3):
.
                    o . . + . . o
                    . . . + . . .
                    . . . + . . .
                   -+-+-+-+-+-+-+-
                    . . . + . . .
                    . . . + . . .
                    o . . + . . o
.
Similarly, for n = 5, we have 4 points with integer coordinates; the point in the first quadrant is at (4,4):
.
                  o . . . + . . . o
                  . . . . + . . . .
                  . . . . + . . . .
                  . . . . + . . . .
                 -+-+-+-+-+-+-+-+-+-
                  . . . . + . . . .
                  . . . . + . . . .
                  . . . . + . . . .
                  o . . . + . . . o
.
For n = 6, we have 12 points, of which the 3 points in the first quadrant are at (4,5), (5,4), and (5,5):
.
                o o . . . + . . . o o
                o . . . . + . . . . o
                . . . . . + . . . . .
                . . . . . + . . . . .
                . . . . . + . . . . .
               -+-+-+-+-+-+-+-+-+-+-+-
                . . . . . + . . . . .
                . . . . . + . . . . .
                . . . . . + . . . . .
                o . . . . + . . . . o
                o o . . . + . . . o o
		

Links

• Kirill Ustyantsev, illustrated example

Crossrefs

Cf. A000328, A016754, A302829, A303642, A303646.

Programs

PARI

a(n) = sum(x=-n+1, n-1, sum(y=-n+1, n-1, (x^2+y^2) > n^2)); \\ Michel Marcus, May 22 2018

Python

import math
for n in range(1, 100):
    count = 0
    for x in range(1, n):
        for y in range(1, n):
            if x * x + y * y > n * n and x < n and y < n:
                count = count + 1
    print(4 * count, end=", ")

Formula

a(n) = A016754(n-1) - A000328(n) - 4.