Tetrahedral number

A tetrahedral number is a number that according to the formula

Can be calculated from a natural number. The first tetrahedral numbers are

For some authors, the zero is not a tetrahedron number, so the number sequence only begins with one.

The name tetrahedral number is derived from a geometric property. If one stone into a tetrahedron by superimposing triangles whose side lengths down each increase by one from the top, then corresponds to the number of stones a tetrahedral number. Here, the number of triangles, and thus also the number of the blocks that form an edge of the tetrahedron. Because of this relationship with a geometric figure include the tetrahedral numbers to the figurate numbers, which include the triangular numbers and square numbers belong. Besides triangles, other polygons can be used as floor plans of the pyramids. This body lead to further pyramid numbers.

Their geometric representation is a tetrahedral clusters in the densest packing of spheres, as they are seen about as decorative layering of oranges (or other spherical fruits) during fruit dealer.

In particular, corresponding to the 20 (the fourth tetrahedral number, represented by a tetrahedral cluster of base length 4) the three-dimensional extension of the Tetraktys (the holy for the Pythagoreans fourth triangular number 10) and includes this as a base and side surfaces.

Noteworthy is a surprising feature of this cluster: Unlike the regular tetrahedron, it is possible with Tetreader clusters to the base of length 4, to fill the space in cubic close packing of spheres gaps.

The formula for the -th tetrahedral number can also be written using a binomial coefficients:

Relations to other figurate numbers

Th the tetrahedron numbers is the sum of the first triangular numbers.

Since the number -th triangle itself is the sum of the first natural numbers, the numbers are the tetrahedral spatial generalization.

Only five numbers are both triangular number and tetrahedral numbers: 1, 10, 120, 1540, 7140 (follow- A027568 in OEIS ).

Three numbers are also square numbers and tetrahedral numbers: 1, 4, 19600th

The tetrahedral numbers can be even sum up again, and the sum of the first tetrahedral numbers is the -th Pentatopzahl.

Sum of the reciprocals

The sum of the reciprocal values ​​of all the tetrahedra numbers.

Generating function

The function

Contains in its series expansion (right side of the equation ) represent the -th tetrahedral number as the coefficient to. Why it is called the generating function of the tetrahedral numbers.