Pyramidal number

In mathematics pyramid numbers or Pyramidalzahlen are a class of Polyederzahlen, that is three-dimensional figured numbers. By some authors, the term Pyramidalzahl is used for the special case of the quadratic Pyramidalzahlen. They are the three-dimensional generalizations of the planar Polygonalzahlen.

Calculation

The each -te- square Pyramidalzahl can be represented by the formula

Calculate.

Alternatively, the -te- square Pyramidalzahl calculated as the sum of the first - squared Polygonalzahlen.

Pyramid numbers to polygons with few vertices

Note: Some authors do not include the zero as the zeroth or first figured number each to other.

More connections with other figurate numbers

The -th square Pyramidalzahl can also be made ​​of the -th triangular number and the -th tetrahedral number according to the formula

Calculate.

In addition, the square Pyramidalzahlen are the sum of two consecutive tetrahedral numbers.

The sum of the first tetrahedral numbers yields a Pentatopzahl, a four-dimensional Figured number.