cp's OEIS Frontend

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

%I A244847 #42 Mar 19 2025 09:04:11
%S A244847 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 A244847 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 A244847 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 A244847 Decimal expansion of rho_c = (5-sqrt(5))/10, the asymptotic critical density for the hard hexagon model.
%C A244847 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
%C A244847 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
%D A244847 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.2 The Golden Mean, phi, p. 7.
%D A244847 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.12.1 Phase transitions in Lattice Gas Models, p. 347.
%H A244847 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 A244847 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 A244847 Alfréd Rényi, <a href="https://static.renyi.hu/renyi_cikkek/1957_representations_for_real_numbers_and_their_ergodic_properties.pdf">Representations for real numbers and their ergodic properties</a>, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.
%H A244847 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/HardHexagonEntropyConstant.html">Hard Hexagon Entropy Constant</a>.
%H A244847 Wikipedia, <a href="http://en.wikipedia.org/wiki/Hard_hexagon_model">Hard Hexagon Model</a>.
%H A244847 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.
%F A244847 Equals 1/(sqrt(5)*phi), where phi = (1+sqrt(5))/2 = A001622. - _Vaclav Kotesovec_, Nov 13 2014
%F A244847 Equals lim_{n -> infinity} A000045(n)/A000032(n+1). - _Bruno Berselli_, Jan 22 2018
%F A244847 Equals Sum_{n>=1} A000045(3^(n-1))/A000032(3^n) = Sum_{n>=1} A045529(n-1)/A006267(n). - _Amiram Eldar_, Dec 20 2018
%F A244847 Equals 1 - A242671. - _Amiram Eldar_, Mar 18 2025
%e A244847 0.2763932022500210303590826331268723764559381640388474275729102754589479...
%t A244847 RealDigits[(5 - Sqrt[5])/10, 10, 102] // First
%Y A244847 Cf. A242671, A244593.
%Y A244847 Cf. A000032, A000045.
%Y A244847 Essentially the same sequence of digits as A229760 and A187799.
%K A244847 nonn,cons,easy
%O A244847 0,1
%A A244847 _Jean-François Alcover_, Nov 12 2014