A019913 - OEIS

%I #38 Aug 21 2023 12:17:52

%S 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,

%T 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,

%U 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

%N Decimal expansion of tangent of 15 degrees.

%C 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

%C 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

%C 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

%C A quadratic integer. - _Charles R Greathouse IV_, Aug 27 2017

%C 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

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

%H Ivan Panchenko, <a href="/A019913/b019913.txt">Table of n, a(n) for n = 0..1000</a>

%H Willis F. Kern and James R. Bland, <a href="https://archive.org/details/in.ernet.dli.2015.205959/page/n99/mode/2up">Solid Mensuration: With Proofs</a>, 2nd ed., J. Wiley & Sons, Inc., New York, 1938. See pp. 91-92.

%H George Albert Wentworth, <a href="https://archive.org/details/newplanespherica00went/page/240/mode/2up">New Plane and Spherical Trigonometry, Surveying, and Navigation</a>, Boston: The Atheneum Press (1903), p. 240.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Exact_trigonometric_constants">Exact trigonometric constants</a>.

%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>

%F Equals Sum_{k>=1} binomial(2*k,k)/(6^k*(k+1)). - _Amiram Eldar_, Jul 11 2020

%F Equals exp(-arccosh(2)). - _Amiram Eldar_, Jul 06 2023

%e 0.2679491924311227064725536...

%t RealDigits[N[Tan[15 Degree], 200]][[1]] (* _Arkadiusz Wesolowski_, Nov 29 2011 *)

%o (PARI) 2-sqrt(3) \\ _Charles R Greathouse IV_, Aug 27 2017

%Y Cf. A002194 (sqrt(3)).

%K nonn,cons

%O 0,1

%A _N. J. A. Sloane_.