A244847 - OEIS

%I #38 Feb 16 2025 08:33:23

%S 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,

%T 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,

%U 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

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

%C 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

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

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.12.1 Phase transitions in Lattice Gas Models, p. 347.

%H Rodney J. Baxter <a href="http://dx.doi.org/10.1088/0305-4470/13/3/007">Hard hexagons: exact solution</a>, Journal of Physics A: Mathematical and General, Vol. 13, No. 3 (1980), pp. L61-L70, <a href="http://yaroslavvb.com/papers/baxter-hard.pdf">alternative link</a>.

%H P. S. Bruckman and I. J. Good, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-3/bruckman.pdf">A Generalization of a Series of De Morgan with Applications of Fibonacci Type</a>, The Fibonacci Quarterly, Vol. 14, No. 3 (1976), pp. 193-196.

%H Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/HardHexagonEntropyConstant.html">Hard Hexagon Entropy Constant</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hard_hexagon_model">Hard Hexagon Model</a>.

%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.

%F Equals 1/(sqrt(5)*phi), where phi = (1+sqrt(5))/2 = A001622. - _Vaclav Kotesovec_, Nov 13 2014

%F Equals lim_{n -> infinity} A000045(n)/A000032(n+1). - _Bruno Berselli_, Jan 22 2018

%F Equals Sum_{n>=1} A000045(3^(n-1))/A000032(3^n) = Sum_{n>=1} A045529(n-1)/A006267(n). - _Amiram Eldar_, Dec 20 2018

%e 0.2763932022500210303590826331268723764559381640388474275729102754589479...

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

%Y Cf. A244593.

%Y Cf. A000032, A000045.

%Y Essentially the same sequence of digits as A229760 and A187799.

%K nonn,cons,easy,changed

%O 0,1

%A _Jean-François Alcover_, Nov 12 2014