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A179260
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Decimal expansion of the connective constant of the honeycomb lattice.
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16
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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
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1,2
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COMMENTS
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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
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]
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)
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REFERENCES
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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).
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LINKS
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FORMULA
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sqrt(2+sqrt(2)) = (2/1)(6/7)(10/9)(14/15)(18/17)(22/23)... (see Sondow-Yi 2010).
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
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EXAMPLE
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1.84775906502257351225636637879357657364483325172728497223019546256107001500...
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MATHEMATICA
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RealDigits[Sqrt[2+Sqrt[2]], 10, 120][[1]] (* Harvey P. Dale, Jan 19 2014 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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