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Beraha Constants

The $n$th Beraha constant is given by

{\rm Be}_n\equiv 2+2\cos\left({2\pi\over n}\right).

The first few are
$\displaystyle {\rm Be}_1$ $\textstyle =$ $\displaystyle 4$  
$\displaystyle {\rm Be}_2$ $\textstyle =$ $\displaystyle 0$  
$\displaystyle {\rm Be}_3$ $\textstyle =$ $\displaystyle 1$  
$\displaystyle {\rm Be}_4$ $\textstyle =$ $\displaystyle 2$  
$\displaystyle {\rm Be}_5$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(3+\sqrt{5}\,)\approx 2.618$  
$\displaystyle {\rm Be}_6$ $\textstyle =$ $\displaystyle 3$  
$\displaystyle {\rm Be}_7$ $\textstyle =$ $\displaystyle 2+2\cos({\textstyle{2\over 7}}\pi) \approx 3.247.$  

They appear to be Roots of the Chromatic Polynomials of planar triangular Graphs. ${\rm Be}_5$ is $\phi+1$, where $\phi$ is the Golden Ratio, and ${\rm Be}_7$ is the Silver Constant.


Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 143, 1983.

© 1996-9 Eric W. Weisstein