# centered hexagonal number

A centered hexagonal number, or hex number is a figurate number^{} that represents a hexagon with a dot in the center and all other dots surrounding the center dot equidistantly. The centered hexagonal number for $n$ is given by the formula $1+6\left(\frac{1}{2}n(n+1)\right)$. In other words, the centered hexagonal number for $n$ is the triangular number^{} for $n$ multiplied by 6, then add 1.

The first few centered hexagonal numbers are: 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, … listed in A003215 of Sloane’s OEIS.

To find centered hexagonal numbers besides 1 that are also triangular numbers or squares, it is necessary to solve Diophantine equations^{}. By solving the Diophantine equation $\frac{1}{2}m(m+1)=3{n}^{2}+3n+1,$ we learn that 91, 8911 and 873181 are numbers that are both centered hexagonal numbers and triangular numbers (they grow very large after that), while solving the Diophantine equation ${m}^{2}=3{n}^{2}+3n+1,$ we learn that 169 and 32761 are centered hexagonal numbers that are also squares.

The sum of the first $n$ centered hexagonal numbers is ${n}^{3}$. The difference between ${(2n)}^{2}$ and the $n$th centered hexagonal number is a number of the form ${n}^{2}+3n-1$, while the difference between $(2n-1)){}^{2}$ and the $n$th centered hexagonal number is an oblong number.

Title | centered hexagonal number |
---|---|

Canonical name | CenteredHexagonalNumber |

Date of creation | 2013-03-22 16:34:35 |

Last modified on | 2013-03-22 16:34:35 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 5 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11D09 |

Synonym | hex number |