A019913 Decimal expansion of tangent of 15 degrees.

2, 6, 7, 9, 4, 9, 1, 9, 2, 4, 3, 1, 1, 2, 2, 7, 0, 6, 4, 7, 2, 5, 5, 3, 6, 5, 8, 4, 9, 4, 1, 2, 7, 6, 3, 3, 0, 5, 7, 1, 9, 4, 7, 4, 6, 1, 8, 9, 6, 1, 9, 3, 7, 1, 9, 4, 4, 1, 9, 3, 0, 2, 0, 5, 4, 8, 0, 6, 6, 9, 8, 3, 0, 9, 1, 1, 9, 9, 9, 6, 2, 9, 1, 8, 8, 5, 3, 8, 1, 3, 2, 4, 2, 7, 5, 1, 4, 2, 4

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• changed

Comments

Also, 2 - sqrt(3) = cotangent of 75 degrees. An equivalent definition of this sequence: decimal expansion of x < 1 satisfying x^2 - 4*x + 1 = 0. - Arkadiusz Wesolowski, Nov 29 2011

Multiplied by -1 (that is, -2 + sqrt(3)), this is one of three real solutions to x^3 = 15x + 4. The other two are 4 and -2 - sqrt(3), all of which can be found with Viete's formula. - Alonso del Arte, Dec 15 2012

Wentworth (1903) shows how to compute the tangent of 15 degrees to five decimal places by the laborious process of adding up the first few terms of Pi/12 + Pi^3/5184 + 2Pi^5/3732480 + 17Pi^7/11287019520 + ... - Alonso del Arte, Mar 13 2015

A quadratic integer. - Charles R Greathouse IV, Aug 27 2017

This is the radius of the largest sphere that can be placed in the space between a sphere of radius 1 and the corners of its circumscribing cube. - Amiram Eldar, Jul 11 2020

Examples

			0.2679491924311227064725536...
		

References

• Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1). Princeton, New Jersey: Princeton University Press (1988): 22 - 23.

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Willis F. Kern and James R. Bland, Solid Mensuration: With Proofs, 2nd ed., J. Wiley & Sons, Inc., New York, 1938. See pp. 91-92.
• George Albert Wentworth, New Plane and Spherical Trigonometry, Surveying, and Navigation, Boston: The Atheneum Press (1903), p. 240.
• Wikipedia, Exact trigonometric constants.
• Index entries for algebraic numbers, degree 2

Crossrefs

Cf. A002194 (sqrt(3)).

Programs

Mathematica

RealDigits[N[Tan[15 Degree], 200]][[1]] (* Arkadiusz Wesolowski, Nov 29 2011 *)

PARI

2-sqrt(3) \\ Charles R Greathouse IV, Aug 27 2017

Formula

Equals Sum_{k>=1} binomial(2*k,k)/(6^k*(k+1)). - Amiram Eldar, Jul 11 2020

Equals exp(-arccosh(2)). - Amiram Eldar, Jul 06 2023

tan(Pi/12) = A019824 / A019884. - R. J. Mathar, Aug 31 2025