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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, 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 18 01:39 EDT 2021. Contains 347504 sequences. (Running on oeis4.)