Pentatope number

A Pentatopzahl is a number that according to the formula

Can be calculated from a natural number. The first Pentatopzahlen are

For some authors, the zero is not Pentatopzahl, so the numbers only begin with the one.

The name Pentatopzahl is derived from the geometric figure of the Pentatops. However, the Pentatop is a four-dimensional body and therefore beyond our imagination. If you were to build a Pentatop the side length evenly balls, so would the number of balls that you need to identical to a Pentatopzahl.

Because of this relationship with a geometric figure include the Pentatopzahlen to the figurate numbers, which also include the square numbers and cubic numbers belong.

Regular figured numbers

Among the regular figurate numbers include:

• Two-dimensional: triangular numbers

The triangle -th number is the sum of the first natural numbers:

• Concepts: Tetrahedron numbers

Th the tetrahedron number is the sum of the first triangular numbers:

Pentatopzahlen

The next regular figurate numbers are then the Pentatopzahlen

As the sum of the first tetrahedral numbers.

Properties

• In the sequence of four numbers Pentatopzahlen are alternately odd and even.
• All regular figurate numbers are in Pascal's triangle. Regular usage applies to the -th Pentatopzahl:

• The series of reciprocals is convergent: The following applies:

• The generating function of the Pentatopzahlen is: