# tetrahedral number

An integer of the form

$$\frac{({n}^{2}+n)(n+2)}{6},$$ |

where $n$ is a nonnegative integer. Sometimes referred to as ${T}_{n}$, tetrahedral numbers^{} are listed in A000292 of Sloane’s OEIS. $2|{T}_{n}$ except when $n\equiv 1mod4$.

With ${t}_{n}$ the $n$th triangular number^{}, the $n$th tetrahedral number can be calculated with this formula:

$${T}_{n}=\sum _{i=1}^{n}{t}_{i}.$$ |

Another way to calculate tetrahedral numbers is with the binomial coefficient^{}

$${T}_{n}=\left(\genfrac{}{}{0pt}{}{n+2}{3}\right).$$ |

This means that tetrahedral numbers can be looked up in Pascal’s triangle.

Tetrahedral numbers have practical applications in sphere packing.

Title | tetrahedral number |
---|---|

Canonical name | TetrahedralNumber |

Date of creation | 2013-03-22 15:56:34 |

Last modified on | 2013-03-22 15:56:34 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 7 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A99 |

Synonym | triangular pyramidal number |