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A168229 Decimal expansion of arctan(sqrt(7)). 5
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 triple 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(4*x) = (cos x)^2. - 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

G. C. Greubel, Table of n, a(n) for n = 1..5000

Kunle Adegoke, Infinite arctangent sums involving Fibonacci and Lucas numbers, arXiv:1603.08097 [math.NT], 2016.

Djurdje Cvijovic, A dilogarithmic integral arising in quantum field theory, arXiv:0911.3773 [math.CA], 2009.

FORMULA

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

Equals Sum_{k >= 1} atan(5*sqrt(7)*F(4k-1)/L(2*(4k-1))) where L=A000032 and F=A000045. See also A033891. - Michel Marcus, Mar 29 2016

EXAMPLE

arctan(sqrt(7)) = 1.209429202888189... .

MATHEMATICA

RealDigits[ArcTan[Sqrt[7]], 10, 50][[1]] (* G. C. Greubel, Nov 18 2017 *)

PROG

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

(MAGMA) [Arctan(Sqrt(7))]; // G. C. Greubel, Nov 18 2017

CROSSREFS

Cf. A195699.

Sequence in context: A189963 A156649 A197330 * A019693 A007493 A262177

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 April 18 05:11 EDT 2021. Contains 343072 sequences. (Running on oeis4.)