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

Author

Jean-François Alcover, Nov 12 2014

Keywords

• nonn
• cons
• easy

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

Examples

			0.2763932022500210303590826331268723764559381640388474275729102754589479...
		

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

• 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.
• 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.

Crossrefs

Cf. A242671, A244593.

Cf. A000032, A000045.

Essentially the same sequence of digits as A229760 and A187799.

Programs

Mathematica

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

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