A111336 Number of convex regular polytopes with n hyperfaces (n>2) or n vertices.

1, 1, 1, 2, 2, 3, 2, 4, 2, 3, 2, 4, 2, 3, 2, 4, 2, 3, 2, 4, 2, 3, 2, 4, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2

Offset

1,4

Comments

The minimum number of hyperfaces required for a convex polytope is 3, and the only convex polytope with 3 hyperfaces is a triangle. - Jianing Song, Sep 17 2018

From Rajan Murthy, Apr 08 2022: (Start)

For n = 1,2,3 point(s) can only be arranged in (n-1)-dimensional simplices. a(1)=a(2)=a(3) = 1. For n = 1,2,3 the point(s) can only be arranged in (n-1)-dimensional simplices. a(1)=a(2)=a(3) = 1.

For higher n, the figures based on n vertices are duals of the figures based on n hyperfaces. (End)

Links

Table of n, a(n) for n=1..89.

Wikipedia, Lista dei politopi regolari.

Wikipedia, Regular Polytopes.

Formula

a(3) = 1; a(n) = 2 if n = 4 or n is odd and >= 5; a(n) = 4 if n = 12, 20, 24, 120, 600 or a power of 2 >= 8; a(n) = 3 otherwise. - Jianing Song, Sep 17 2018

Example

a(8) = 4 because the regular polytopes with 8 faces are the octagon, the octahedron, the four-dimensional cube and the 7-dimensional simplex.
From Rajan Murthy, Apr 08 2022: (Start)
For n = 8, points may be arranged in an octagon, a cube, a 4-dimensional orthoplex, or a 7-dimensional simplex, so a(8) = 4.
For n = 12, there are a(12) = 4 regular polytopes with 12 hyperfaces. They, and their duals with 12 points, are:
  12 hyperfaces        12 points
   dodecagon            dodecagon
   dodecahedron         icosahedron
   6-cube               6-D orthoplex
   11-D simplex         11-D simplex
(End)

Prog

(PARI)
a(n)={
    if(n<=3, return(1));
    if(n==4||(n>=5&&n%2==1), return(2));
    if(n>=6&&n%2==0, return(3+(n==12||n==20||n==24||n==120||n==600||(n>=8&&omega(2*n)==1))));
    else(return(0));
} \\ Jianing Song, Sep 17 2018

Crossrefs

Cf. A302139 (indices of 4).

Sequence in context: A038148 A366742 A141829 * A083902 A205562 A217984

Adjacent sequences: A111333 A111334 A111335 * A111337 A111338 A111339

Keyword

nonn,easy

Author

Paulo de A. Sachs (sachs6(AT)yahoo.de), Nov 09 2005

Extensions

Terms beyond a(38) from Jianing Song, Sep 17 2018

a(1) and a(2) prepended and definition extended by Rajan Murthy, Apr 08 2022

Status

approved