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A244593
Decimal expansion of z_c = phi^5 (where phi is the golden ratio), a lattice statistics constant which is the exact value of the critical activity of the hard hexagon model.
4
1, 1, 0, 9, 0, 1, 6, 9, 9, 4, 3, 7, 4, 9, 4, 7, 4, 2, 4, 1, 0, 2, 2, 9, 3, 4, 1, 7, 1, 8, 2, 8, 1, 9, 0, 5, 8, 8, 6, 0, 1, 5, 4, 5, 8, 9, 9, 0, 2, 8, 8, 1, 4, 3, 1, 0, 6, 7, 7, 2, 4, 3, 1, 1, 3, 5, 2, 6, 3, 0, 2, 3, 1, 4, 0, 9, 4, 5, 1, 2, 2, 4, 8, 5, 3, 6, 0, 3, 6, 0, 2, 0, 9, 4, 6, 9, 5, 5, 6, 8, 7, 4, 2
OFFSET
2,4
COMMENTS
Essentially the same digit sequence as A239798, A019863 and A019827. - R. J. Mathar, Jul 03 2014
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
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved