Tetrahedral number

A tetrahedral cluster of base length 5 comprising 35 spheres. Each floor represents one of the five first triangular numbers .

A tetrahedral number is a number that is based on the formula

${\displaystyle {\frac {n(n+1)(n+2)}{6}}}$

can be calculated from a natural number . The first tetrahedral numbers are ${\displaystyle n}$

0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... (sequence A000292 in OEIS )

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

The name tetrahedral number is derived from a geometric property. If you put stones into a tetrahedron by laying triangles on top of each other, the side lengths of which increase by one from top to bottom, then the number of stones corresponds to a tetrahedron number. This is the number of these triangles and thus also the number of stones that form an edge of the tetrahedron. Because of this relationship with a geometric figure , the tetrahedral numbers are among the figured numbers , which also include the triangular and square numbers . In addition to triangles, other polygons can also be used as the outline of pyramids. These bodies lead to further pyramid numbers . ${\displaystyle n}$

Their geometric representation is a tetrahedral cluster in the closest packing of spheres , as can be seen as a decorative layering of oranges (or other spherical fruits) at the fruit dealer.

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

A surprising property of these clusters is remarkable: In contrast to the regular tetrahedron, with tetrahedron clusters up to a base length of 4 it is possible to fill the space in the cubic closest packing of spheres without any gaps.

The formula for the -th tetrahedral number can also be written using a binomial coefficient : ${\displaystyle n}$

${\displaystyle T_{n}={\binom {n+2}{3}}.}$

Relationships to other figured numbers

The -th tetrahedral number is the sum of the first triangular numbers . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \Delta _{i}}$

${\displaystyle T_{n}=\sum _{i=1}^{n}\Delta _{i}=\Delta _{1}+\Delta _{2}+\ldots +\Delta _{n}}$

Since the -th triangular number is itself the sum of the first natural numbers, the tetrahedral numbers are their spatial generalizations. ${\displaystyle n}$ ${\displaystyle n}$

Only five numbers are both triangle number and tetrahedron number: 1, 10, 120, 1540, 7140. (Sequence A027568 in OEIS )

Three numbers are both a square and a tetrahedron: 1, 4, 19600.

The tetrahedral numbers can be summed up again and the sum of the first tetrahedral numbers is the -th pentatope number . ${\displaystyle n}$ ${\displaystyle n}$

Sum of the reciprocal values

The sum of the reciprocal values ​​of all tetrahedral numbers is . ${\displaystyle {\tfrac {3}{2}}}$

${\displaystyle \sum _{n=1}^{\infty }T_{n}^{-1}=\sum _{n=1}^{\infty }{\frac {6}{n(n+1)(n+2)}}={\frac {3}{2}}}$

Generating function

The function

${\displaystyle {\frac {x}{(x-1)^{4}}}=1x+4x^{2}+10x^{3}+20x^{4}+\ldots }$

contains in its series expansion (right side of the equation) the -th tetrahedral number as a coefficient to . It is therefore called the generating function of the tetrahedral numbers. ${\displaystyle n}$ ${\displaystyle x^{n}}$

Web links

• Eric W. Weisstein : Tetrahedral number . In: MathWorld (English).
• Cubically closest packing of spheres