Polygonal number

A Polygonalzahl is a number for which there is a regular polygon (polygon ) that can be set with a corresponding number of stones. 16 For example, a Polygonalzahl, since a square can be set from 16 stones. The Polygonalzahlen include the triangular and square numbers.

The Polygonalzahlen among the figurate numbers. Another way to recycle numbers on polygons represent the centered Polygonalzahlen

The Polygonalzahlen can be generated by a simple calculation rule. One chooses to difference as a natural number. The first number is always 1, and any subsequent Polygonalzahlen caused by the difference in each case added to the preceding one. The following examples illustrate this.

The individual summands are the result of terms of an arithmetic sequence with initial term 1 and the respective difference ( cf. difference sequence ), respectively. This structure of the Polygonalzahlen is also reflected in the corresponding polygons:

The 16 is the fourth square number.

22 is the fourth Fünfeckszahl.

The 28 is the fourth Sechseckszahl.

Occasionally, as the 0-th triangular number, square number, ... is introduced by definition. According to this convention is the result of the triangular numbers, for example.

Calculation

The each -te- Eckszahl can be represented by the formula

Calculate.

If there is any - Eckszahl, then calculates the associated according to the formula

Derivation

Be the number of pages. The n-th k- Eckzahl that is formed in that pages will be expanded by 1 point. The flared sides have common points. Thus, the (n 1 )-th k- Eckzahl has points more than the n th k- Eckszahl. The nth k- Eckszahl is therefore:

To I.: Application of the Gaussian sum formula

Sum of the reciprocals

The sum of the reciprocals of each all - Eckszahlen is convergent. The following applies:

Applications

After Fermat Polygonalzahlensatz each number can be represented as a sum of at most - Eckszahlen.