A105230
Number of n-dimensional polytopes with vertices from {0,1}^n.
Original entry on oeis.org
1, 5, 151, 60879, 4292660729, 18446743888401503325
Offset: 1
A105232
Number of n-dimensional polytopes with vertices from {0,1}^n up to combinatorial equivalence.
Original entry on oeis.org
1, 2, 8, 192, 1050136
Offset: 1
A276412
Number of 0/1 n-simplices formed from vertices of unit n-dimensional cube, including degenerate ones.
Original entry on oeis.org
1, 1, 6, 27, 472, 19735, 2773763, 1245930065, 1816451773537, 8687099045170277, 138409456269288832810, 7456837714994642537616273, 1376366010506271334134714585969, 880266808159237325569935079783591707
Offset: 1
A126776
Basis orbits of n-dimensional cubes.
Original entry on oeis.org
1, 1, 4, 17, 237, 9892
Offset: 1
A140800
Total number of vertices in all finite n-dimensional convex regular polytopes, or 0 if the number is infinite.
Original entry on oeis.org
1, 2, 0, 50, 773, 48, 83, 150, 281, 540, 1055, 2082, 4133, 8232, 16427, 32814, 65585, 131124, 262199, 524346, 1048637, 2097216, 4194371, 8388678, 16777289, 33554508, 67108943, 134217810, 268435541, 536871000, 1073741915, 2147483742
Offset: 0
a(0) = 1 because the 0-D regular polytope is the point.
a(1) = 2 because the only regular 1-D polytope is the line segment, with 2 vertices, one at each end.
a(2) = 0, indicating infinity, because the regular k-gon has k vertices.
a(3) = 50 (4 for the tetrahedron, 6 for the octahedron, 8 for the cube, 12 for the icosahedron, 20 for the dodecahedron) = the sum of A053016.
a(4) = 773 = 5 + 8 + 16 + 24 + 120 + 600 = sum of A063924.
For n>4 there are only the three regular n-dimensional polytopes, the simplex with n+1 vertices, the hypercube with 2^n vertices and the hyperoctahedron = cross polytope = orthoplex with 2*n vertices, for a total of A086653(n) + 1 = 2^n + 3*n + 1 (again restricted to n > 4).
- H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973.
- Branko Grunbaum, Convex Polytopes, second edition (first edition (1967) written with the cooperation of V. L. Klee, M. Perles and G. C. Shephard; second edition (2003) prepared by V. Kaibel, V. L. Klee and G. M. Ziegler), Graduate Texts in Mathematics, Vol. 221, Springer 2003.
- P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, Vol. 92, Cambridge University Press, Cambridge, 2002.
Cf.
A000943,
A000944,
A019503,
A053016,
A060296,
A063924,
A063925,
A063926,
A063927,
A065984,
A086653,
A093478,
A093479,
A105230,
A105231.
-
LinearRecurrence[{4, -5, 2}, {1, 2, 0, 50, 773, 48, 83, 150}, 32] (* Georg Fischer, May 03 2019 *)
Showing 1-5 of 5 results.
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