The th Beraha constant (or number) is given by
is , where is the golden ratio, is the silver constant, and . The following table summarizes the first few Beraha numbers.
Noninteger Beraha numbers can never be roots of any chromatic polynomials with the possible exception of (G. Royle, pers. comm., Nov. 21, 2005). However, the roots of chromatic polynomials of planar triangulations appear to cluster around the Beraha numbers (and, technically, are conjectured to be accumulation points of roots of planar triangulation chromatic polynomials).