A303743 a(n) is a number of lattice points in 3D Cartesian grid between cube with edge length 2*n centered in origin and its inscribed sphere. Three pairs of the cube's faces are parallel to the planes XOY, XOZ, YOZ respectively.

0, 0, 8, 92, 220, 412, 784, 1272, 1848, 2696, 3692, 5020, 6460, 8176, 10248, 12720, 15464, 18476, 21988, 25924, 30016, 35040, 40248, 46052, 52388, 59132, 66364, 74416, 83256, 92304, 102500, 112988, 124076, 136252, 148936, 162648, 176928, 192332, 208100, 225284, 243088

Offset: 1

Author

Kirill Ustyantsev, Apr 29 2018

Keywords

• nonn

Comments

If two parallel faces of the inscribed cube are parallel XOY-plane and other two pairs are parallel planes x=y and x=-y respectively we'll have another sequence.

Examples

			For n=3 we have 8 points between the defined cube and its inscribed sphere:
  (-2,-2,-2)
  (-2,-2, 2)
  (-2, 2,-2)
  (-2, 2, 2)
  ( 2,-2,-2)
  ( 2,-2, 2)
  ( 2, 2,-2)
  ( 2, 2, 2)
		

Crossrefs

Cf. A000605, A016755.

For the 2D case see A303642.

Programs

PARI

a(n) = sum(x=-n+1, n-1, sum(y=-n+1, n-1, sum(z=-n+1, n-1, x*x+y*y+z*z>n^2))); \\ Michel Marcus, Jun 23 2018

Python

for n in range (1, 42):
  count=0
  n2 = n*n
  for x in range(-n+1, n):
    for y in range(-n+1, n):
      for z in range(-n+1, n):
        if x*x+y*y+z*z > n2:
          count += 1
  print(count)

Formula

a(n) = A016755(n-1) - A000605(n) - 6.