A244847 Decimal expansion of rho_c = (5-sqrt(5))/10, the asymptotic critical density for the hard hexagon model.

2, 7, 6, 3, 9, 3, 2, 0, 2, 2, 5, 0, 0, 2, 1, 0, 3, 0, 3, 5, 9, 0, 8, 2, 6, 3, 3, 1, 2, 6, 8, 7, 2, 3, 7, 6, 4, 5, 5, 9, 3, 8, 1, 6, 4, 0, 3, 8, 8, 4, 7, 4, 2, 7, 5, 7, 2, 9, 1, 0, 2, 7, 5, 4, 5, 8, 9, 4, 7, 9, 0, 7, 4, 3, 6, 2, 1, 9, 5, 1, 0, 0, 5, 8, 5, 5, 8, 5, 5, 9, 1, 6, 2, 1, 2, 1, 7, 7, 2, 5, 0, 3

Offset

0,1

Comments

The vertical distance between the accumulation point and the outermost point of a golden spiral inscribed inside a golden rectangle with dimensions phi and 1 along the x and y axes, respectively (the horizontal distance is A176015). - Amiram Eldar, May 18 2021

The limiting frequency of the digit 1 in the base phi representation of real numbers in the range [0,1], where phi is the golden ratio (A001622) (Rényi, 1957). - Amiram Eldar, Mar 18 2025

This constant gives the linear asymptotic coefficient of the average number of basepairs in an RNA secondary structure on a number n of vertices tending to infinity (see Theorem 3 in Bu, Kauers, and Zeilberger). - Stefano Spezia, Feb 27 2026

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.2 The Golden Mean, phi, p. 7.

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=0..101.

Rodney J. Baxter, Hard hexagons: exact solution, Journal of Physics A: Mathematical and General, Vol. 13, No. 3 (1980), pp. L61-L70, alternative link.

P. S. Bruckman and I. J. Good, A Generalization of a Series of De Morgan with Applications of Fibonacci Type, The Fibonacci Quarterly, Vol. 14, No. 3 (1976), pp. 193-196.

AJ Bu, Manuel Kauers, and Doron Zeilberger, Statistical Analysis of Hairpins and BasePairs in RNA Secondary Structures, arXiv:2602.19255 [math.CO], 2026.

Hideyuki Ohtsuka, Problem H-922, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 61, No. 3 (2023), p. 282; Solution to Problem Problem H-922, by Yunyong Zhang, ibid., Vol. 63, No. 1 (2025), pp. 126-127.

Alfréd Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.

Eric Weisstein's MathWorld, Hard Hexagon Entropy Constant.

Wikipedia, Hard Hexagon Model.

Index entries for algebraic numbers, degree 2.

Formula

Equals 1/(sqrt(5)*phi), where phi = (1+sqrt(5))/2 = A001622. - Vaclav Kotesovec, Nov 13 2014

Equals lim_{n -> infinity} A000045(n)/A000032(n+1). - Bruno Berselli, Jan 22 2018

Equals Sum_{n>=1} A000045(3^(n-1))/A000032(3^n) = Sum_{n>=1} A045529(n-1)/A006267(n). - Amiram Eldar, Dec 20 2018

Equals 1 - A242671. - Amiram Eldar, Mar 18 2025

Equals Sum_{n>=1} (4^n/A000032(2^n)^2)*Sum_{k=1..n} A000045(2^k)/4^k (Ohtsuka, 2023). - Amiram Eldar, Dec 16 2025

Example

0.2763932022500210303590826331268723764559381640388474275729102754589479...

Mathematica

RealDigits[(5 - Sqrt[5])/10, 10, 102] // First

Crossrefs

Cf. A242671, A244593.

Cf. A000032, A000045.

Essentially the same sequence of digits as A229760 and A187799.

Sequence in context: A350761 A135155 A308705 * A175477 A389421 A333521

Adjacent sequences: A244844 A244845 A244846 * A244848 A244849 A244850

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Nov 12 2014

Status

approved