# Sequence: total number of vertices in all n-dimensional regular polytopes

Jonathan Post jvospost3 at gmail.com
Mon Jul 14 18:52:23 CEST 2008

```Total number of vertices in all n-dimensional regular polytopes, or -1
if the number is infinite.

n  a(n)
0   1
1   2
2   -1
3   50
4   773
5   47
6   82
7  149
8  280
9  539
10 1054
11 2081
12 4132
13 8231
14 16426
15 32813
16 65584
17 131123
18 262198
19 524345
20 1048636
21 2097215
22 4194370
23 8388677
24 16777288
25 33554507
26 67108942
27 134217809
28 268435540
29 536870999
30 1073741914
31 2147483741

Examples:
a(0) = 1 because the 0-D regular poltope 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) = 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 vertices, the hypercube with 2^n vertices, and the
hyperoctahedron = cross polytope = orthoplex with 2*n vertices, for a
total of A086653  2^n + 3*n (again, restricted to n>4).

Comment:
There are a prime number of vertices of the full set of n-dimensional
regular polytopes beginning: a(4) = 773, a(5) = 47, a(7) = 149, a(11)
= 2081, a(13) = 8231, a(47) = 140737488355469.
This sequence is also, for the higher n-th dimensions, the total
number of (n-1)-dimensional faces (facets).

Cf. A053016, A060296, A063924-A063927, A086653

X-SIG5: cfc9b7817e66b0e96e0ee5a925ae4960

<p><br /><br />Thanks for your message.</p><br /><p><br />Best regards.<br />Washington .</p>

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