

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
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OFFSET

1,3


COMMENTS

If two parallel faces of the inscribed cube are parallel XOYplane 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(n1)  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
..for x in range (n, n):
...for y in range (n, n):
....for z in range (n, n):
.....if (x*x+y*y+z*z>n*n and x>n and x<n and y>n and y<n and z>n and z<n):
......count=count+1
.print(count)
(PARI) a(n) = sum(x=n+1, n1, sum(y=n+1, n1, sum(z=n+1, n1, 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



