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

2,4

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.

Links

Table of n, a(n) for n=2..104.

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

((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

Cf. A019863, A019827, A085850, A239798.

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

Sequence in context: A197070 A197333 A226120 * A335415 A277524 A118811

Adjacent sequences: A244590 A244591 A244592 * A244594 A244595 A244596

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jul 01 2014

Status

approved