OFFSET
2,4
COMMENTS
The minimal polynomial of this constant is x^2 - 11*x - 1. - Joerg Arndt, Jan 01 201
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.12.1 Phase transitions in Lattice Gas Models, p. 347.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 83.
LINKS
Eric Weisstein's MathWorld, Hard Hexagon Entropy Constant.
D. W. Wood and R. W. Turnbull, z^2-11z-1 as an algebraic invariant for the hard-hexagon model, 1988 J. Phys. A: Math. Gen. 21 L989.
Wikipedia, Hard Hexagon Model.
FORMULA
Equals ((1 + sqrt(5))/2)^5 = (11 + 5*sqrt(5))/2.
Equals phi^5 = 11 + 1/phi^5 = 3 + 5*phi, an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Nov 11 2023
Equals lim_{n->infinity} S(n, 5*(-1 + 2*phi))/ S(n-1, 5*(-1 + 2*phi)), with the S-Chebyshev polynomials (see A049310). Wolfdieter Lang, Nov 15 2023
EXAMPLE
11.09016994374947424102293417182819058860154589902881431067724311352630...
MATHEMATICA
RealDigits[GoldenRatio^5, 10, 103] // First
PROG
(PARI) (5*sqrt(5)+11)/2 \\ Charles R Greathouse IV, Aug 10 2016
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Jul 01 2014
STATUS
approved