Английская Википедия:Beraha constants

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Шаблон:Short description The Beraha constants are a series of mathematical constants by which the <math>n\text{th}</math> Beraha constant is given by

<math>B (n) = 2 + 2 \cos \left ( \frac{2\pi}{n} \right ).</math>

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

The following table summarizes the first ten Beraha constants.

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

See also

• Chromatic polynomial

References

• Шаблон:Cite web
• 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 <math>\lambda = 1</math>," Research Report COPR 72–7, University of Waterloo, 1972a.
• Tutte, W. T. "Chromatic Sums for Planar Triangulations IV: The Case <math>\lambda = \infty</math>." Research Report COPR 72–4, University of Waterloo, 1972b.

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