A300074 Decimal expansion of 1/(2*sin(Pi/5)) = A121570/2.

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

Offset: 0

Author

Wolfdieter Lang, Mar 01 2018

Keywords

• nonn
• cons
• easy

Comments

This is the reciprocal of A182007, and one half of A121570.

This is the ratio of the radius r of the circumscribing circle of a regular pentagon and its side length s: r/s = 1/(2*sin(Pi/5)).

A quartic number of denominator 5 and minimal polynomial 5x^4 - 5x^2 + 1. - Charles R Greathouse IV, Mar 04 2018

Appears at Schur decomposition of A=[1 2; 2 3]. - Donghwi Park, Jun 20 2018

Examples

			r/s = 0.850650808352039932181540497063011072240401403764816881836740242377...
2*r/s = A121570.
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..2000
• Wikipedia, Schur decomposition.
• Index entries for algebraic numbers, degree 4.

Crossrefs

Cf. A001622, A090773, A121570, A182007, A195693, A195723.

Programs

Mathematica

RealDigits[1/(2 Sin[Pi/5]), 10, 111][[1]] (* Robert G. Wilson v, Jul 15 2018 *)

PARI

1/(2*sin(Pi/5)) \\ Charles R Greathouse IV, Mar 04 2018

PARI

sqrt((5+sqrt(5))/10) \\ Charles R Greathouse IV, Mar 04 2018

Formula

r/s = 1/A182007 = A121570/2 = (2*phi - 1)*sqrt(2 + phi)/5, with the golden ratio phi = (1 + sqrt(5))/2 = A001622.

From Amiram Eldar, Feb 08 2022: (Start)

Equals cos(arccot(phi)) = cos(arctan(1/phi)) = cos(A195693).

Equals sin(arctan(phi)) = sin(arccot(1/phi)) = sin(A195723). (End)

Equals Product_{k>=1} (1 + (-1)^k/A090773(k)). - Amiram Eldar, Nov 23 2024