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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.

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