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

$${\rm Be}_1 = 4$$

$${\rm Be}_2 = 0$$

$${\rm Be}_3 = 1$$

$${\rm Be}_4 = 2$$

$${\rm Be}_5 = {\textstyle{1\over 2}}(3+\sqrt{5}\,)\approx 2.618$$

$${\rm Be}_6 = 3$$

$${\rm Be}_7 = 2+2\cos({\textstyle{2\over 7}}\pi) \approx 3.247.$$

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.

References

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