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A168229 Decimal expansion of arctan(sqrt(7)). 3
1, 2, 0, 9, 4, 2, 9, 2, 0, 2, 8, 8, 8, 1, 8, 8, 8, 1, 3, 6, 4, 2, 1, 3, 3, 0, 1, 5, 3, 1, 9, 0, 8, 4, 7, 6, 1, 0, 8, 5, 9, 7, 5, 4, 5, 6, 4, 7, 5, 3, 3, 2, 7, 7, 6, 6, 7, 4, 0, 9, 5, 2, 2, 9, 8, 6, 2, 0, 5, 4, 5, 1, 2, 1, 8, 5, 7, 8, 9, 3, 6, 6, 8, 3, 1, 6, 0, 3, 6, 0, 7, 2, 0, 1, 5, 0, 7, 8, 8, 2, 1, 4, 6, 0, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This constant is used on p.6 of Cvijovic. Recently, an interesting dilogarithmic integral arising in quantum field theory has been closed-form evaluated in terms of the Clausen function Cl_2(theta) by Coffey [J. Math. Phys.} 49 (2008), 093508]. It represents the volume of an ideal tetrahedron in hyperbolic space and is involved in two intriguing equivalent conjectures of Borwein and Broadhurst. It is shown here, by simple and direct arguments, that this integral can be expressed by the triplet of the Clausen function values which are involved in one of the two above-mentioned conjectures.

This constant is the least x>0 satisfying cos(4x)=(cos x)^2.  [From Clark Kimberling, Oct 15 2011]

An identity resembling Machin's Pi/4 = arctan(1/1) = 4*arctan(1/5) - arctan(1/239) is  arctan(sqrt(7)/1) = 5*arctan(sqrt(7)/11) + 2*arctan(sqrt(7)/181), which can also be expressed as arcsin(sqrt(7/2^3)) = 5*arcsin(sqrt(7/2^7)) + 2*arcsin(sqrt(7/2^15)) (cf. A038198).  [Joerg Arndt, Nov 09 2012]

LINKS

Table of n, a(n) for n=1..105.

Djurdje Cvijovic, A dilogarithmic integral arising in quantum field theory, Nov 29, 2009.

FORMULA

Smallest positive solution of cos(x) + sqrt(1 + cos^2(x)) = sqrt(2). - Geoffrey Caveney, Apr 24 2014

EXAMPLE

1.209429202888189.

PROG

(PARI) atan(sqrt(7)) \\ Michel Marcus, Mar 11 2013

CROSSREFS

Sequence in context: A189963 A156649 A197330 * A019693 A007493 A136319

Adjacent sequences:  A168226 A168227 A168228 * A168230 A168231 A168232

KEYWORD

cons,nonn

AUTHOR

Jonathan Vos Post, Nov 20 2009

EXTENSIONS

More digits from R. J. Mathar, Dec 06 2009

STATUS

approved

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Last modified October 31 10:58 EDT 2014. Contains 248861 sequences.