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.

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

Author

Jean-François Alcover, Jul 01 2014

Keywords

• nonn
• cons
• easy

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 2017

Examples

			11.09016994374947424102293417182819058860154589902881431067724311352630...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.12.1 Phase transitions in Lattice Gas Models, p. 347.
• Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
• 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.
• Wikipedia, Hard Hexagon Model.
• 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.

Crossrefs

Cf. A019863, A019827, A085850, A239798.

Cf. A001622, A014176, A098316, A098317, A098318, A176398, A176439, A176458 A176522, A176537 (metallic means).

Programs

Mathematica

RealDigits[GoldenRatio^5, 10, 103] // First

PARI

(5*sqrt(5)+11)/2 \\ Charles R Greathouse IV, Aug 10 2016

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