This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
A 2 X 2 X 2 grid of points defines 28 distinct lines which intersect in a total of 15 points (including the original 8 points).
L[0]=0; L1[1]=0; R1[1]=0;
L[n_]:=L[n]=2*L1[n]-L[n-1]+R1[n]
L1[n_]:=L1[n]=2*L[n-1]-L1[n-1]+R2[n]
R1[n_]:=R1[n]=R1[n-1]+4*(EulerPhi[n-1]-e[n])
e[n_]:=If[Mod[n,2]==0,0,EulerPhi[(n-1)/2]]
R2[n_]:= If[Mod[n,2]==0,(n-1)*EulerPhi[n-1], If[Mod[n,4]==1,(n-1)*EulerPhi[n-1]/2,0]]
Table[L[n],{n,0,37}] (* Seppo Mustonen, Apr 25 2009 *)
The 8 points of the 2 X 2 X 2 grid define 20 planes: the 6 faces of the cube, 6 planes formed by an edge and its opposite edge, and 8 planes defined by the three vertices surrounding a given vertex. So, a(2)=20.
The 20 planes defined by the points of the 2 X 2 X 2 grid fall into 13 families of parallel planes: 3 parallel to the faces, 4 perpendicular to the space diagonals and 6 which are the planes defined by an edge and its opposite edge. So a(2)=13.
A 2 X 2 X 2 grid of points defines 28 distinct lines which intersect in a total of 15 points (including the original 8 points). One of these points is the center of the grid, 8 of them are the vertices of the cube and 6 are the face-centers of the cube. So, a(2) = 3.
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