Hexagonal number

Nested hexagons made up of 28 spheres

A hexagonal number or hexagonal number is a number created using the formula

${\displaystyle n\cdot (2n-1)=2n^{2}-n}$

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

0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, ... (sequence A000384 in OEIS )

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

The name hexagon number is derived from the geometric figure of the hexagon . Hexagonal numbers can be laid out as nested hexagons with increasing edge length. With the triangular numbers and square numbers, they belong to the class of polygonal numbers . If you form the hexagons around a common center, then you get the hex numbers (also called centered hexagon numbers).

Definition and calculation

Hexagonal numbers can be interpreted as a hexagon , as the following figure shows:

The -th hexagonal number is sometimes referred to as . It can also be calculated using the -th triangular number. ${\displaystyle n}$ ${\displaystyle S_{n}}$ ${\displaystyle (n-1)}$ ${\displaystyle \Delta _{n-1}}$

${\displaystyle S_{n}=n+4\cdot \Delta _{n-1}}$

Sums of hexagonal numbers

According to Fermat's theorem of polygons, each number can be represented as the sum of six hexagonal numbers. Adrien-Marie Legendre proved in 1830 that any number greater than 1792 can even be written as the sum of just four hexagonal numbers. Among the smaller numbers there are 13 numbers that are not the sum of four hexagonal numbers:

5, 10, 11, 20, 25, 26, 38, 39, 54, 65, 70, 114, 130 (sequence A007527 in OEIS ).

Of these, 11 and 26 are the only numbers that cannot be represented as the sum of five hexagonal numbers.

The sum of the first hexagonal numbers is ${\displaystyle k}$

${\displaystyle \sum _{i=1}^{k}S_{i}={\frac {4k^{3}+3k^{2}-k}{6}}}$

the -th hexagonal pyramidal number ${\displaystyle k}$

Series of reciprocals

The sum of the reciprocal values ​​of all hexagonal numbers is

${\displaystyle \sum _{k=1}^{\infty }{S_{k}}^{-1}=\sum _{k=1}^{\infty }{\frac {1}{2\cdot k^{2}-k}}=2\ln {(2)}}$

Individual evidence

1. ^ Leonard Eugene Dickson : History of the Theory of Numbers. Volume 2: Diophantine Analysis. Dover Publications, Mineola NY 2005, ISBN 0-486-44233-0 , p. 18
2. ↑ Eric W. Weisstein : Hexagonal Number . In: MathWorld (English).