A168229 Decimal expansion of arctan(sqrt(7)).

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

Offset

1,2

Comments

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.

Index entries for transcendental numbers

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

Equals arccos(1/(2*sqrt(2))). - Amiram Eldar, May 28 2021

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. A000032, A000045, A033891, A038198, A195699.

Sequence in context: A156649 A197330 A343882 * 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