A338808 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, created when an n-antiprism, formed from two n-sided regular polygons joined by 2n adjacent alternating triangles, is internally cut by all the planes defined by any three of its vertices.

8, 128, 56, 8, 0, 3, 450, 270, 82, 20, 10, 0, 2, 2592, 2376, 972, 204, 168, 48, 0, 0, 5, 7266, 7574, 4550, 2254, 660, 336, 98, 14, 14, 0, 2, 0, 0, 0, 0, 0, 0, 2, 27216, 31088, 15632, 5360, 1904, 432, 128, 0, 0, 0, 0, 0, 9, 68778, 84240, 61272, 33138, 15714, 5400, 1946, 720, 270, 126, 72, 18, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4

Offset

3,1

Comments

See A338806 for further details and images for this sequence.

The author thanks Zach J. Shannon for assistance in producing the images for this sequence.

Links

Table of n, a(n) for n=3..79.

Hyung Taek Ahn and Mikhail Shashkov, Geometric Algorithms for 3D Interface Reconstruction.

Scott R. Shannon, 4-antiprism, showing the 56 5-faced polyhedra. See A338806 for an image of the full polyhedra.

Scott R. Shannon, 4-antiprism, showing the 8 6-faced polyhedra

Scott R. Shannon, 4-antiprism, showing the 3 8-faced polyhedra

Scott R. Shannon, 7-antiprism, showing the 7266 4-faced polyhedra. See A338806 for an image of the full polyhedra.

Scott R. Shannon, 7-antiprism, showing the 7574 5-faced polyhedra

Scott R. Shannon, 7-antiprism, showing the 4550 6-faced polyhedra

Scott R. Shannon, 7-antiprism, showing the 2254 7-faced polyhedra

Scott R. Shannon, 7-antiprism, showing the 660 8-faced polyhedra

Scott R. Shannon, 7-antiprism, showing the 336 9-faced polyhedra.

Scott R. Shannon, 7-antiprism, showing the 98 10-faced polyhedra. None of these are visible on the surface.

Scott R. Shannon, 7-antiprism, showing the 14 11-faced, 14 12-faced, 2 14-faced, 2 21-faced polyhedra. These are colored white, black, red, yellow respectively. None of these are visible on the surface.

Scott R. Shannon, 10-antiprism, showing the 13 20-faced polyhedra. See A338806 for an image of the full polyhedra.

Eric Weisstein's World of Mathematics, Antiprism.

Wikipedia, Antiprism.

Formula

Sum of row n = A338806(n).

Example

The 4-antiprism is cut with 16 internal planes defined by all 3-vertex combinations of its 8 vertices. This leads to the creation of 128 4-faced polyhedra, 56 5-faced polyhedra, 8 6-faced polyhedra, and 3 8-faced polyhedra, 195 pieces in all. Note the number of 8-faced polyhedra is not a multiple of 4 - they lie directly along the z-axis so need not be a multiple of the number of edges forming the regular n-gons.
The table begins:
8;
128,56,8,0,3;
450,270,82,20,10,0,2;
2592,2376,972,204,168,48,0,0,5;
7266,7574,4550,2254,660,336,98,14,14,0,2,0,0,0,0,0,0,2;
27216,31088,15632,5360,1904,432,128,0,0,0,0,0,9;
68778,84240,61272,33138,15714,5400,1946,720,270,126,72,18,0,0,4,0,0,0,0,0,0,0,0,4;
194580,235880,153620,68580,25240,7460,2560,660,200,0,0,0,0,0,0,0,13;

Crossrefs

Cf. A338806 (number of polyhedra), A338801 (regular prism), A338622 (Platonic solids), A333543 (n-dimensional cube).

Sequence in context: A034220 A034239 A171755 * A363330 A233089 A204194

Adjacent sequences: A338805 A338806 A338807 * A338809 A338810 A338811

Keyword

nonn,tabf

Author

Scott R. Shannon, Nov 10 2020

Status

approved