A333543 Irregular triangle read by rows: T(n,k) (n >= 1, k >= n+1) is the number of cells with k vertices in the dissection of an n-dimensional cube by all the hyperplanes that pass through any n vertices.

1, 4, 72, 24, 162816, 96576, 118464, 64896, 45888, 22464, 19776, 11904, 8640, 8448, 6144, 1728, 1152, 384, 384, 384

Offset

1,2

Comments

Rows 1 through 4 computed by Veit Elser, later confirmed by Tom Karzes.

The row sums give A333539.

References

N. J. A. Sloane, Cutting Up a Cube, Math Fun Mailing List, Apr 10 2020; with replies from Tom Karzes, Tomas Rokicki, Veit Elser, and others.

Links

Table of n, a(n) for n=1..20.

Veit Elser, Rows 1 through 4

Scott R. Shannon, Illustration for a(2) = 4.

Scott R. Shannon, Illustration for a(3) = 72. This shows the 4-faced cells in the 3D cube dissection. The 72 pieces have been moved away from the origin a distance proportional to the average distance of their vertices from the origin.

Scott R. Shannon, Illustration for a(4) = 24. This shows the 5-faced cells in the 3D cube dissection. The 24 pieces have been moved away from the origin a distance proportional to the average distance of their vertices from the origin. These polyhedra form a perfect octahedron inside the original cube with its points touching the cube's inner surface.

Example

The two diagonals of a square cut it into four triangular pieces, so T(2,3) = 4.
Triangle begins:
1,
4,
72, 24,
162816, 96576, 118464, 64896, 45888, 22464, 19776, 11904, 8640, 8448, 6144, 1728, 1152, 384, 384, 384,
...

Crossrefs

Cf. A333539, A333540, A333544, A338622 (number of k-faced polyhedra for the 3D Platonic solids).

For the number of hyperplanes see A007847.

Sequence in context: A354859 A088693 A322397 * A262235 A133003 A358293

Adjacent sequences: A333540 A333541 A333542 * A333544 A333545 A333546

Keyword

nonn,tabf,more

Author

N. J. A. Sloane, Apr 21 2020

Status

approved