Centered hexagon number

37 spheres in the form of nested hexagons

A centered hexagon number or hex number is a number that is based on the formula

${\displaystyle 3n^{2}-3n+1\,}$

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

1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, ... (follow A003215 in OEIS )

A centered hexagon number indicates a number of circles, so that a circle in the middle is so evenly surrounded by circles that they form a regular hexagon . They belong to the centered polygonal numbers , thus also to the figured numbers .

Relationships to other figured numbers

Cube numbers

The sum of the first centered hexagon numbers results in the -th cube number : ${\displaystyle n}$ ${\displaystyle ZS_{i}}$ ${\displaystyle n}$ ${\displaystyle K_{n}}$

1 = 1; 1 + 7 = 8; 1 + 7 + 19 = 27; 1 + 7 + 19 + 37 = 64; ...

${\displaystyle K_{n}=ZS_{1}+ZS_{2}+\ldots +ZS_{n}}$

Square numbers

If you have the equation

${\displaystyle m^{2}=3n^{2}-3n+1\ }$

you can find centered hexagon numbers that are also square numbers, such as 169 and 32761.

Triangle numbers

The -th centered hexagonal number can also be found according to the formula ${\displaystyle n}$

${\displaystyle 6\cdot \Delta _{n-1}+1=6\cdot {\frac {n\cdot (n-1)}{2}}+1}$

calculate using the -th triangular number. ${\displaystyle (n-1)}$ ${\displaystyle \Delta _{n}}$

If you have the equation

${\displaystyle {\frac {m\cdot (m+1)}{2}}=3n^{2}-3n+1\ }$

one can find centered hexagonal numbers that are also triangular numbers, such as: 91, 8911 and 873181.

Sum of the reciprocal values

The sum of the reciprocal values ​​of the centered hexagonal numbers is convergent: It applies

${\displaystyle \sum _{k=1}^{\infty }ZS_{k}^{-1}={\sqrt {3}}\cdot {\frac {-{\sqrt {3}}+\pi \cdot \tanh {\frac {\pi {\sqrt {3}}}{6}}}{3}}=0{,}3053841...}$

Web links

• Eric W. Weisstein : Centered hexagonal number . In: MathWorld (English).