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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.

%I A303743 #28 May 24 2021 07:33:11
%S A303743 0,0,8,92,220,412,784,1272,1848,2696,3692,5020,6460,8176,10248,12720,
%T A303743 15464,18476,21988,25924,30016,35040,40248,46052,52388,59132,66364,
%U A303743 74416,83256,92304,102500,112988,124076,136252,148936,162648,176928,192332,208100,225284,243088
%N 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.
%C A303743 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 A303743 a(n) = A016755(n-1) - A000605(n) - 6.
%e A303743 For n=3 we have 8 points between the defined cube and its inscribed sphere:
%e A303743   (-2,-2,-2)
%e A303743   (-2,-2, 2)
%e A303743   (-2, 2,-2)
%e A303743   (-2, 2, 2)
%e A303743   ( 2,-2,-2)
%e A303743   ( 2,-2, 2)
%e A303743   ( 2, 2,-2)
%e A303743   ( 2, 2, 2)
%o A303743 (Python)
%o A303743 for n in range (1, 42):
%o A303743   count=0
%o A303743   n2 = n*n
%o A303743   for x in range(-n+1, n):
%o A303743     for y in range(-n+1, n):
%o A303743       for z in range(-n+1, n):
%o A303743         if x*x+y*y+z*z > n2:
%o A303743           count += 1
%o A303743   print(count)
%o A303743 (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 A303743 Cf. A000605, A016755.
%Y A303743 For the 2D case see A303642.
%K A303743 nonn
%O A303743 1,3
%A A303743 _Kirill Ustyantsev_, Apr 29 2018