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

1,2

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

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.

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)

Example

1.376381920471173538...

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 *)

Prog

(PARI) tan(3*Pi/10) \\ Charles R Greathouse IV, Aug 27 2017
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Tan(3*Pi(R)/10); // G. C. Greubel, Nov 22 2018
(Sage) numerical_approx(tan(3*pi/10), digits=100) # G. C. Greubel, Nov 22 2018
(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

Crossrefs

Cf. A019845, A019863, A019934.

Cf. A344171 (rhombic triacontahedron surface area).

Cf. A344172 (rhombic triacontahedron volume).

Cf. A344212 (rhombic triacontahedron midradius).

Sequence in context: A301814 A065281 A256848 * A157699 A283270 A159779

Adjacent sequences: A019949 A019950 A019951 * A019953 A019954 A019955

Keyword

nonn,cons

Author

N. J. A. Sloane

Status

approved