

A179260


Decimal expansion of the connective constant of the honeycomb lattice.


12



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
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OFFSET

1,2


COMMENTS

This is the case n=8 of the ratio Gamma(1/n)*Gamma((n1)/n)/(Gamma(2/n)*Gamma((n2)/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 radius of the circumscribed circle. This ratio is 2 for the longest diagonal.  Wolfdieter Lang, May 11 2017
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. 6974. 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. 6974.
N. Madras and G. Slade, Selfavoiding walks, Probability and its Applications. Birkhäuser Boston, Inc. Boston, MA (1993).


LINKS

Table of n, a(n) for n=1..100.
Hugo DuminilCopin, Stanislav Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt2), arXiv:1007.0575 [mathph], 2011.
Hugo DuminilCopin, Stanislav Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt2), Ann. Math. 175 (2012), pp. 16531665.
S. Finch, Errata and Addenda to Mathematical Constants, Jun 23 2012, Section 5.10
Steven R. Finch, Errata and Addenda to Mathematical Constants, January 22, 2016. [Cached copy, with permission of the author]
G. Lawler, O. Schramm and W. Werner, On the scaling limit of planar selfavoiding walk, Fractal Geometry and applications: a jubilee of Benoit Mandelbrot, Part 2, 339364. Proc.
B. Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, Phys. Rev. Lett. 49 (1982), 10621065.
Jonathan Sondow and Huang Yi, New Wallis and Catalantype infinite products for Pi, e, and sqrt(2+sqrt(2)), arXiv:1005.2712 [math.NT], 2010.
Jonathan Sondow and Huang Yi, New Wallis and Catalantype infinite products for Pi, e, and sqrt(2+sqrt(2)), Amer. Math. Monthly 117 (2010) 912917.
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 SondowYi 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


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.
Sequence in context: A249415 A021122 A110233 * A254246 A254375 A019684
Adjacent sequences: A179257 A179258 A179259 * A179261 A179262 A179263


KEYWORD

cons,nonn


AUTHOR

Jonathan Vos Post, Jul 06 2010


STATUS

approved



