Pentatope number
Pentatope numbers (hypertenrahedral numbers) represent a 4-dimensional generalization of 2-dimensional triangular numbers ; analogous to the 3-dimensional tetrahedral numbers . Due to their relationship to a geometric figure , the pentatope numbers are figured numbers .
The pentatope number is calculated as follows:
The first pentatop numbers are: (0), 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, ... (sequence A000332 in OEIS )
designation
The name pentatop number is derived from the geometric figure of the pentatope . It is a four-dimensional body that emerges from a three-dimensional tetrahedron. If a pentatope of the same length could be composed of (hyper) spheres, their number would be identical to a pentatope number.
Regular figured numbers
The regular figured numbers include:
- Two-dimensional: triangular numbers
The -th triangular number is the sum of the first natural numbers :
- Three-dimensional: tetrahedral numbers
The -th tetrahedral number is the sum of the first triangular numbers:
- Four-dimensional: pentatope numbers
The next regular figured numbers are the pentatope numbers, they are created by summing the first tetrahedral numbers :
properties
- In the sequence of pentatopes, four numbers are alternating between odd and even.
- All regular figured numbers are in Pascal's triangle , the pentatop numbers are on the fifth diagonal. In particular, the following applies to the -th pentatope number:
- The series of reciprocal values is convergent:
- The generating function of the pentatope numbers is: