Hexagonal number

A Sechseckszahl or Hexagonalzahl is a number according to the formula

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

For some authors, the zero is not Sechseckszahl, so that the sequence of numbers begins with the first one.

The name Sechseckszahl is derived from the geometric figure of the hexagon. Sechseckszahlen can be set as a nested hexagons with increasing edge length. They count the triangle numbers and square numbers to the class of Polygonalzahlen. If one forms the hexagons around a common center, then you get the hex numbers (also called centered Sechseckszahlen ).

Sechseckszahlen can be designed as a hexagon, as the following figure shows:

The - te Sechseckszahl is called partially. It can also be calculated with the help of the -th triangular number.

Sums of Sechseckszahlen

Each number can be represented as the sum of six Sechseckszahlen after the Fermat Polygonalzahlensatz. Adrien -Marie Legendre 1830 proved that any number, greater than 1792 can be written as a sum of four even Sechseckszahlen. Among the smaller numbers, there are 13 numbers that are not the sum of four Sechseckszahlen:

Of these, 11 and 26 are the only numbers that can not be represented as a sum of five Sechseckszahlen.

Is the sum of the first Sechseckszahlen

The -th hexagonal Pyramidalzahl

Series of the reciprocals

Is the sum of the reciprocal values ​​of all Sechseckszahlen