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 ${\displaystyle \phi +1}$, where ${\displaystyle \phi }$ is the golden ratio, ${\displaystyle B(7)}$is the silver constant^[1] (also known as the silver root),^[2] and ${\displaystyle B(10)=\phi +2}$.

The following table summarizes the first ten Beraha constants.

• Chromatic polynomial

• Weisstein, Eric W.. "Beraha Constants". http://mathworld.wolfram.com/BerahaConstants.html. 
• 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 ${\displaystyle \lambda =1}$," Research Report COPR 72–7, University of Waterloo, 1972a.
• Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case ${\displaystyle \lambda =\mathrm{\infty }}$." Research Report COPR 72–4, University of Waterloo, 1972b.

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