A140800 Total number of vertices in all finite n-dimensional convex regular polytopes, or 0 if the number is infinite.

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

Author

Jonathan Vos Post, Jul 15 2008

Keywords

• easy
• nonn

Comments

Andrew Weimholt suggests a related sequence, namely "total number of vertices in all finite n-dimensional regular polytopes, or 0 if the number is infinite, includes both convex and non-convex, beginning: 1, 2, 0, 106, 2453, 48, 83, 150, 281, 540, ... and writes that the sequence of just the non-convex cases (0, 0, -1, 56, 1680, 0, 0, 0, ..., where "-1" indicates infinity as zero is otherwise employed) is not as interesting, since it's all zeros from a(5) on.

Examples

			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).
		

References

• 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.

Links

• Georg Fischer, Table of n, a(n) for n = 0..1000
• Gil Kalai, Five Open Problems Regarding Convex Polytopes.
• Eric W. Weisstein, Polytope
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A000943, A000944, A019503, A053016, A060296, A063924, A063925, A063926, A063927, A065984, A086653, A093478, A093479, A105230, A105231.

Programs

Mathematica

LinearRecurrence[{4, -5, 2}, {1, 2, 0, 50, 773, 48, 83, 150}, 32] (* Georg Fischer, May 03 2019 *)

Formula

For n > 4, a(n) = A086653(n) + 1 = 2^n + 3*n + 1.

G.f.: -(1488*x^7 - 3656*x^6 + 2794*x^5 - 569*x^4 - 58*x^3 + 3*x^2 + 2*x - 1)/((1-x)^2*(1-2*x)). [Colin Barker, Sep 05 2012]

Extensions

a(14)-a(15) corrected by Georg Fischer, May 02 2019