A303743 - OEIS

%I #28 May 24 2021 07:33:11

%S 0,0,8,92,220,412,784,1272,1848,2696,3692,5020,6460,8176,10248,12720,

%T 15464,18476,21988,25924,30016,35040,40248,46052,52388,59132,66364,

%U 74416,83256,92304,102500,112988,124076,136252,148936,162648,176928,192332,208100,225284,243088

%N 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.

%C 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.

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

%e For n=3 we have 8 points between the defined cube and its inscribed sphere:

%e   (-2,-2,-2)

%e   (-2,-2, 2)

%e   (-2, 2,-2)

%e   (-2, 2, 2)

%e   ( 2,-2,-2)

%e   ( 2,-2, 2)

%e   ( 2, 2,-2)

%e   ( 2, 2, 2)

%o (Python)

%o for n in range (1, 42):

%o   count=0

%o   n2 = n*n

%o   for x in range(-n+1, n):

%o     for y in range(-n+1, n):

%o       for z in range(-n+1, n):

%o         if x*x+y*y+z*z > n2:

%o           count += 1

%o   print(count)

%o (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

%Y Cf. A000605, A016755.

%Y For the 2D case see A303642.

%K nonn

%O 1,3

%A _Kirill Ustyantsev_, Apr 29 2018