A019952 Decimal expansion of tangent of 54 degrees.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

Also the decimal expansion of cotangent of 36 degrees. - Mohammad K. Azarian, Jun 30 2013

A quartic number with denominator 5. - Charles R Greathouse IV, Aug 27 2017

Conjecture: Product (2/3) * (8/7) * (12/13) * (18/17) * (22/23) * (32/33) * ... * (a_n/b_n) = sqrt(25 + 10*sqrt(5))/5 = tan(3*Pi/10) = A019952, where a_n even, a_n + b_n = a(n), |a_n - b_n| = 1, n >= 0. - Dimitris Valianatos, Feb 14 2020

Also the limiting value of the distance between the lines F(n)*x + F(n+1)*y = 0 and F(n)*x + F(n+1)*y = F(n+2) (where F(n)=A000045(n) are the Fibonacci numbers and n>0). - Burak Muslu, Apr 03 2021

Decimal expansion of the radius of an inscribed sphere in a rhombic triacontahedron with unit edge length. - Wesley Ivan Hurt, May 11 2021

Examples

			1.376381920471173538207209581910887679525899336...
		

Links

• Ivan Panchenko, Table of n, a(n) for n = 1..1000
• Wikipedia, Exact trigonometric constants.
• Wikipedia, Rhombic triacontahedron.
• Index entries for algebraic numbers, degree 4.

Crossrefs

Cf. A019845, A019863, A019934, A090772, A375067.

Cf. A344171 (rhombic triacontahedron surface area).

Cf. A344172 (rhombic triacontahedron volume).

Cf. A344212 (rhombic triacontahedron midradius).

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); Tan(3*Pi(R)/10); // G. C. Greubel, Nov 22 2018

Maple

Digits:=100: evalf(tan(3*Pi/10)); # Wesley Ivan Hurt, Oct 07 2014

Mathematica

RealDigits[Tan[3*Pi/10], 10, 100][[1]] (* Wesley Ivan Hurt, Oct 07 2014 *)
RealDigits[Tan[54 Degree],10,120][[1]] (* Harvey P. Dale, Jul 16 2016 *)

PARI

tan(3*Pi/10) \\ Charles R Greathouse IV, Aug 27 2017

Python

from sympy import sqrt
[print(i, end=', ') for i in str(sqrt(1+2/sqrt(5)).n(110)) if i!='.'] # Karl V. Keller, Jr., Jun 19 2020

Sage

numerical_approx(tan(3*pi/10), digits=100) # G. C. Greubel, Nov 22 2018

Formula

Equals A019863/A019845 = 1/A019934. - R. J. Mathar, Jul 26 2010

The largest positive solution of cos(4*arctan(1/x)) = cos(6*arctan(1/x)). - Thomas Olson, Oct 03 2014

Equals sqrt(25 + 10*sqrt(5))/5. - G. C. Greubel, Nov 22 2018

Equals sqrt(2 + sqrt(5))/5^(1/4). - Burak Muslu, Apr 03 2021

From Wesley Ivan Hurt, May 11 2021: (Start)

Equals phi^2/sqrt(1+phi^2) where phi is the golden ratio.

Equals sqrt(1+2/sqrt(5)). (End)

Equals Product_{k>=1} (1 - (-1)^k/A090772(k)). - Amiram Eldar, Nov 23 2024

Equals 2*A375067. - Hugo Pfoertner, Nov 23 2024