Beraha constants

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

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

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

The following table summarizes the first ten Beraha constants.

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

See also

• Chromatic polynomial

References

• Weisstein, Eric W. "Beraha Constants". Wolfram MathWorld. Retrieved November 3, 2018.
• Beraha, S. Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1974.
• Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 143, 1983.
• Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, pp. 160–163, 1986.
• Tutte, W. T. "Chromials." University of Waterloo, 1971.
• Tutte, W. T. "More about Chromatic Polynomials and the Golden Ratio." In Combinatorial Structures and their Applications: Proc. Calgary Internat. Conf., Calgary, Alberta, 1969. New York: Gordon and Breach, p. 439, 1969.
• Tutte, W. T. "Chromatic Sums for Planar Triangulations I: The Case $\lambda =1$," Research Report COPR 72–7, University of Waterloo, 1972a.
• Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case $\lambda =\infty$." Research Report COPR 72–4, University of Waterloo, 1972b.