Beraha constants

The Beraha constants are a series of mathematical constants by which the ${\displaystyle n{\text{th}}}$ Beraha constant is given by

${\displaystyle B(n)=2+2\cos \left({\frac {2\pi }{n}}\right).}$

Notable examples of Beraha constants include ${\displaystyle B(5)}$ is ${\displaystyle \varphi +1}$, where ${\displaystyle \varphi }$ is the golden ratio, ${\displaystyle B(7)}$ is the silver constant^[1] (also known as the silver root),^[2] and ${\displaystyle B(10)=\varphi +2}$.

The following table summarizes the first ten Beraha constants.

${\displaystyle n}$	${\displaystyle B(n)}$	Approximately
1	4
2	0
3	1
4	2
5	${\displaystyle {\frac {1}{2}}(3+{\sqrt {5}})}$	2.618
6	3
7	${\displaystyle 2+2\cos({\tfrac {2}{7}}\pi )}$	3.247
8	${\displaystyle 2+{\sqrt {2}}}$	3.414
9	${\displaystyle 2+2\cos({\tfrac {2}{9}}\pi )}$	3.532
10	${\displaystyle {\frac {1}{2}}(5+{\sqrt {5}})}$	3.618