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

1,3

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.

Links

Table of n, a(n) for n=1..41.

Formula

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

Example

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)

Prog

(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)
(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

Crossrefs

Cf. A000605, A016755.

For the 2D case see A303642.

Sequence in context: A298013 A302614 A220573 * A187157 A332597 A331448

Adjacent sequences: A303740 A303741 A303742 * A303744 A303745 A303746

Keyword

nonn

Author

Kirill Ustyantsev, Apr 29 2018

Status

approved