A105231
Number of n-dimensional polytopes with vertices from {0,1}^n up to (0,1)-equivalence.
Original entry on oeis.org
1, 2, 12, 347, 1226525, 400507800465455
Offset: 1
- Oswin Aichholzer, Extremal Properties of 0/1-Polytopes of Dimension 5. In: Polytopes - Combinatorics and Computation, DMV Seminar vol 29, Birkhäuser, Basel, 2000.
- William Y. C. Chen and Peter L. Guo, Equivalence Classes of Full-Dimensional 0/1-Polytopes with Many Vertices, Discrete Comput. Geom., 52 (2014), 630-662.
- Andreas Paffenholz and the polymake team, polyDB. See Polytopes > Geometric Polytopes > 0/1 Polytopes by Oswin Aichholzer.
- Chuanming Zong, What is known about unit cubes, Bull. Amer. Math. Soc., 42 (2005), 181-211.
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
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.
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LinearRecurrence[{4, -5, 2}, {1, 2, 0, 50, 773, 48, 83, 150}, 32] (* Georg Fischer, May 03 2019 *)
Showing 1-3 of 3 results.
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