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A338801 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, created when an n-prism, formed from two n-sided regular polygons joined by n adjacent rectangles, is internally cut by all the planes defined by any three of its vertices.

%I A338801 #38 Dec 07 2020 01:44:44
%S A338801 17,0,1,72,24,575,450,232,60,15,0,3,1728,1668,948,144,24,12,8799,
%T A338801 10080,6321,3052,898,490,161,14,35,14,7,22688,24080,12784,4160,1248,
%U A338801 272,80,32,78327,101142,70254,39708,19584,6894,2369,1062,351,54,27,18,27,36,11,165500,203220,134860,62520,21240,5720,1080,300,100,20
%N A338801 Irregular table read by rows: The number of k-faced polyhedra, where k>=4, created when an n-prism, formed from two n-sided regular polygons joined by n adjacent rectangles, is internally cut by all the planes defined by any three of its vertices.
%C A338801 See A338783 for further details and images for this sequence.
%C A338801 The author thanks Zach J. Shannon for assistance in producing the images for this sequence.
%H A338801 Hyung Taek Ahn and Mikhail Shashkov, <a href="https://cnls.lanl.gov/~shashkov/papers/ahn_geometry.pdf">Geometric Algorithms for 3D Interface Reconstruction</a>.
%H A338801 Scott R. Shannon, <a href="/A338801/a338801.jpg">3-prism, showing the 18 polyhedra post-cutting and exploded</a>. Each piece has been moved away from the origin by a distance proportional to the average distance of its vertices from the origin. Red shows the seventeen 4-faced polyhedra, orange the single 6-faced polyhedron.
%H A338801 Scott R. Shannon, <a href="/A338801/a338801_1.jpg">7-prism, showing the 8799 4-faced polyhedra</a>.
%H A338801 Scott R. Shannon, <a href="/A338801/a338801_2.jpg">7-prism, showing the 10080 5-faced polyhedra</a>.
%H A338801 Scott R. Shannon, <a href="/A338801/a338801_3.jpg">7-prism, showing the 6321 6-faced polyhedra</a>.
%H A338801 Scott R. Shannon, <a href="/A338801/a338801_4.jpg">7-prism, showing the 3052 7-faced polyhedra</a>.
%H A338801 Scott R. Shannon, <a href="/A338801/a338801_5.jpg">7-prism, showing the 898 8-faced polyhedra</a>.
%H A338801 Scott R. Shannon, <a href="/A338801/a338801_6.jpg">7-prism, showing the 490 9-faced polyhedra</a>.
%H A338801 Scott R. Shannon, <a href="/A338801/a338801_7.jpg">7-prism, showing the 161 10-faced polyhedra</a>.
%H A338801 Scott R. Shannon, <a href="/A338801/a338801_8.jpg">7-prism, showing the 14 11-faced, 35 12-faced, 14 13-faced, 7 14-faced polyhedra</a>. These are colored white, black, yellow, red respectively. None of these are visible on the surface.
%H A338801 Scott R. Shannon, <a href="/A338801/a338801_9.jpg">7-prism, showing all 29871 polyhedra</a>.  The 4,5,6,7,8,9,10 faced polyhedra are colored red, orange, yellow, green, blue, indigo, violet respectively. The 11,12,13,14 faced polyhedra are not visible on the surface.
%H A338801 Scott R. Shannon, <a href="/A338801/a338801_10.jpg">10-prism, showing all 594560 polyhedra</a>.
%H A338801 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prism.html">Prism</a>.
%H A338801 Wikipedia, <a href="https://en.wikipedia.org/wiki/Prism_(geometry)">Prism (geometry)</a>.
%F A338801 Sum of row n = A338783(n).
%e A338801 The triangular 3-prism is cut with 6 internal planes defined by all 3-vertex combinations of its 6 vertices. This leads to the creation of seventeen 4-faced polyhedra and one 6-faced polyhedra, eighteen pieces in all. The single 6-faced polyhedra lies at the very center of the original 3-prism.
%e A338801 The 9-prism is cut with 207 internal planes leading to the creation of 319864 pieces. It is noteworthy in creating all k-faced polyhedra from k=4 to k=18.
%e A338801 The table begins:
%e A338801 17,0,1;
%e A338801 72,24;
%e A338801 575,450,232,60,15,0,3;
%e A338801 1728,1668,948,144,24,12;
%e A338801 8799,10080,6321,3052,898,490,161,14,35,14,7;
%e A338801 22688,24080,12784,4160,1248,272,80,32;
%e A338801 78327,101142,70254,39708,19584,6894,2369,1062,351,54,27,18,27,36,11;
%e A338801 165500,203220,134860,62520,21240,5720,1080,300,100,20;
%Y A338801 Cf. A338783 (number of polyhedra), A338808 (antiprism), A338622 (Platonic solids), A333543 (n-dimensional cube).
%K A338801 nonn,tabf
%O A338801 3,1
%A A338801 _Scott R. Shannon_, Nov 10 2020