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

1,4

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.

Links

Table of n, a(n) for n=1..54.

Kirill Ustyantsev, illustrated example

Formula

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

Example

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

Prog

(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=", ")
(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

Crossrefs

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

Sequence in context: A120033 A097073 A019085 * A298796 A106232 A359709

Adjacent sequences: A303641 A303642 A303643 * A303645 A303646 A303647

Keyword

nonn

Author

Kirill Ustyantsev, Apr 27 2018

Status

approved