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

Author

Jonathan Vos Post, Nov 20 2009

Keywords

• cons
• nonn

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

Examples

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

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

Crossrefs

Cf. A000032, A000045, A033891, A038198, A195699.

Programs

Magma

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

Mathematica

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

PARI

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

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

Extensions

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