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

Author

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

Keywords

• nonn
• easy

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)

Examples

			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)
		

Links

• Wikipedia, Lista dei politopi regolari.
• Wikipedia, Regular Polytopes.

Crossrefs

Cf. A302139 (indices of 4).

Programs

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

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

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