A179260 Decimal expansion of the connective constant of the honeycomb lattice.

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, 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, 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

Offset

1,2

Comments

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

An algebraic integer of degree 4: largest root of x^4 - 4x^2 + 2. - Charles R Greathouse IV, Nov 05 2014

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]

From Wolfdieter Lang, Apr 29 2018: (Start)

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.

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)

References

Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.

N. Madras and G. Slade, Self-avoiding walks, Probability and its Applications. Birkhäuser Boston, Inc. Boston, MA (1993).

Links

Table of n, a(n) for n=1..100.

Hugo Duminil-Copin and Stanislav Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt2), arXiv:1007.0575 [math-ph], 2011.

Hugo Duminil-Copin and Stanislav Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt2), Ann. Math. 175 (2012), pp. 1653-1665.

S. Finch, Errata and Addenda to Mathematical Constants, Jun 23 2012, Section 5.10; arXiv:2001.00578 [math.HO], 2020.

Pierre-Louis Giscard, Que sait-on compter sur un graphe. Partie 3 (in French), Images des Mathématiques, CNRS, 2020.

G. Lawler, O. Schramm and W. Werner, On the scaling limit of planar self-avoiding walk, Fractal Geometry and applications: a jubilee of Benoit Mandelbrot, Part 2, 339-364. Proc.

B. Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, Phys. Rev. Lett. 49 (1982), 1062-1065.

Jonathan Sondow and Huang Yi, New Wallis- and Catalan-type infinite products for Pi, e, and sqrt(2+sqrt(2)), arXiv:1005.2712 [math.NT], 2010.

Jonathan Sondow and Huang Yi, New Wallis- and Catalan-type infinite products for Pi, e, and sqrt(2+sqrt(2)), Amer. Math. Monthly 117 (2010) 912-917.

Index entries for algebraic numbers, degree 4

Index entries for sequences related to Chebyshev polynomials.

Formula

sqrt(2+sqrt(2)) = (2/1)(6/7)(10/9)(14/15)(18/17)(22/23)... (see Sondow-Yi 2010).

Equals 1/A154739. - R. J. Mathar, Jul 11 2010

Equals 2*A144981. - Paul Muljadi, Aug 23 2010

log (A001668(n)) ~ n log k where k = sqrt(2+sqrt(2)). - Charles R Greathouse IV, Nov 08 2013

2*cos(Pi/8) = sqrt(2+sqrt(2)). See a remark on the smallest diagonal in the octagon above. - Wolfdieter Lang, May 11 2017

Equals also 2*sin(3*Pi/8). See the comment on van Roomen's third problem above. - Wolfdieter Lang, Apr 29 2018

Equals i^(1/4) + i^(-1/4). - Gary W. Adamson, Jul 06 2022

Equals Product_{k>=0} ((8*k + 2)*(8*k + 6))/((8*k + 1)*(8*k + 7)). - Antonio Graciá Llorente, Feb 24 2024

Example

1.84775906502257351225636637879357657364483325172728497223019546256107001500...

Mathematica

RealDigits[Sqrt[2+Sqrt[2]], 10, 120][[1]] (* Harvey P. Dale, Jan 19 2014 *)

Prog

(PARI) sqrt(2+sqrt(2)) \\ Charles R Greathouse IV, Nov 05 2014

Crossrefs

Cf. A002193, A101464, A127672, A144981, A154739, A302715, A121601.

Sequence in context: A249415 A021122 A110233 * A254246 A369103 A254375

Adjacent sequences: A179257 A179258 A179259 * A179261 A179262 A179263

Keyword

cons,nonn

Author

Jonathan Vos Post, Jul 06 2010

Status

approved