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%I #82 Feb 25 2024 09:35:59
%S 1,8,4,7,7,5,9,0,6,5,0,2,2,5,7,3,5,1,2,2,5,6,3,6,6,3,7,8,7,9,3,5,7,6,
%T 5,7,3,6,4,4,8,3,3,2,5,1,7,2,7,2,8,4,9,7,2,2,3,0,1,9,5,4,6,2,5,6,1,0,
%U 7,0,0,1,5,0,0,2,2,0,4,7,1,7,4,2,9,6,7,9,8,6,9,7,0,0,6,8,9,1,9,2
%N Decimal expansion of the connective constant of the honeycomb lattice.
%C This is the case n=8 of the ratio Gamma(1/n)*Gamma((n-1)/n)/(Gamma(2/n)*Gamma((n-2)/n)). - _Bruno Berselli_, Dec 13 2012
%C An algebraic integer of degree 4: largest root of x^4 - 4x^2 + 2. - _Charles R Greathouse IV_, Nov 05 2014
%C This number is also the length ratio of the shortest diagonal (not counting the side) of the octagon and the side. This ratio is A121601 for the longest diagonal. - _Wolfdieter Lang_, May 11 2017 [corrected Oct 28 2020]
%C From _Wolfdieter Lang_, Apr 29 2018: (Start)
%C This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète. See the Havil reference, problem 3, pp. 69-74. See also the comments in A302711 with the Romanus link and his Exemplum tertium.
%C This problem is equivalent to R(45, 2*sin(Pi/120)) = 2*sin(3*Pi/8) with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672. For the constant 2*sin(Pi/120) see A302715. (End)
%D Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.
%D N. Madras and G. Slade, Self-avoiding walks, Probability and its Applications. Birkhäuser Boston, Inc. Boston, MA (1993).
%H Hugo Duminil-Copin and Stanislav Smirnov, <a href="http://arxiv.org/abs/1007.0575">The connective constant of the honeycomb lattice equals sqrt(2+sqrt2)</a>, arXiv:1007.0575 [math-ph], 2011.
%H Hugo Duminil-Copin and Stanislav Smirnov, <a href="http://dx.doi.org/10.4007/annals.2012.175.3.14">The connective constant of the honeycomb lattice equals sqrt(2+sqrt2)</a>, Ann. Math. 175 (2012), pp. 1653-1665.
%H S. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, Jun 23 2012, Section 5.10; arXiv:2001.00578 [math.HO], 2020.
%H Pierre-Louis Giscard, <a href="http://images.math.cnrs.fr/Que-sait-on-compter-sur-un-graphe-Partie-3.html">Que sait-on compter sur un graphe. Partie 3</a> (in French), Images des Mathématiques, CNRS, 2020.
%H G. Lawler, O. Schramm and W. Werner, <a href="http://arxiv.org/abs/math/0204277">On the scaling limit of planar self-avoiding walk</a>, Fractal Geometry and applications: a jubilee of Benoit Mandelbrot, Part 2, 339-364. Proc.
%H B. Nienhuis, <a href="http://dx.doi.org/10.1103/PhysRevLett.49.1062">Exact critical point and critical exponents of O(n) models in two dimensions</a>, Phys. Rev. Lett. 49 (1982), 1062-1065.
%H Jonathan Sondow and Huang Yi, <a href="http://arxiv.org/abs/1005.2712">New Wallis- and Catalan-type infinite products for Pi, e, and sqrt(2+sqrt(2))</a>, arXiv:1005.2712 [math.NT], 2010.
%H Jonathan Sondow and Huang Yi, <a href="http://www.jstor.org/stable/10.4169/000298910x523399">New Wallis- and Catalan-type infinite products for Pi, e, and sqrt(2+sqrt(2))</a>, Amer. Math. Monthly 117 (2010) 912-917.
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F sqrt(2+sqrt(2)) = (2/1)(6/7)(10/9)(14/15)(18/17)(22/23)... (see Sondow-Yi 2010).
%F Equals 1/A154739. - _R. J. Mathar_, Jul 11 2010
%F Equals 2*A144981. - _Paul Muljadi_, Aug 23 2010
%F log (A001668(n)) ~ n log k where k = sqrt(2+sqrt(2)). - _Charles R Greathouse IV_, Nov 08 2013
%F 2*cos(Pi/8) = sqrt(2+sqrt(2)). See a remark on the smallest diagonal in the octagon above. - _Wolfdieter Lang_, May 11 2017
%F Equals also 2*sin(3*Pi/8). See the comment on van Roomen's third problem above. - _Wolfdieter Lang_, Apr 29 2018
%F Equals i^(1/4) + i^(-1/4). - _Gary W. Adamson_, Jul 06 2022
%F Equals Product_{k>=0} ((8*k + 2)*(8*k + 6))/((8*k + 1)*(8*k + 7)). - _Antonio Graciá Llorente_, Feb 24 2024
%e 1.84775906502257351225636637879357657364483325172728497223019546256107001500...
%t RealDigits[Sqrt[2+Sqrt[2]],10,120][[1]] (* _Harvey P. Dale_, Jan 19 2014 *)
%o (PARI) sqrt(2+sqrt(2)) \\ _Charles R Greathouse IV_, Nov 05 2014
%Y Cf. A002193, A101464, A127672, A144981, A154739, A302715, A121601.
%K cons,nonn
%O 1,2
%A _Jonathan Vos Post_, Jul 06 2010
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