A020762 Decimal expansion of 1/sqrt(5).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

This number is the cosine of the central angle of a regular icosahedron; see A105199 for the angle itself. - Clark Kimberling, Feb 10 2009

Largest radius of ten circles tangent to a circle of radius 1. - Charles R Greathouse IV, Jan 14 2013

Examples

			0.447213595499957939281834733746255247088123671922305144854179449082104...
		

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Hideyuki Ohtsuka, Problem B-1148, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 52, No. 2 (2014), p. 179; The Exact Value of an Infinite Series, Solution to B-1148, ibid., Vol. 53, No. 2 (2015), pp. 183-184.
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A000032, A000045, A010476, A001566, A058635, A105199.

Programs

Mathematica

RealDigits[5^(-1/2), 10, 150] (* Stefan Steinerberger, Apr 08 2006 *)
Circs[n_] := With[{r = Sin[Pi/n]/(1 - Sin[Pi/n])}, Graphics[Append[
  Table[Circle[(r + 1) {Sin[2 Pi k/n], Cos[2 Pi k/n]}, r], {k, n}],
    {Blue, Circle[{0, 0}, 1]}]]]
Circs[10] (* Charles R Greathouse IV, Jan 14 2013 *)

PARI

1/sqrt(5) \\ Charles R Greathouse IV, Jan 14 2013

Formula

Equals cos(arctan(2)). - Clark Kimberling, Feb 10 2009

Equals lim_{n -> infinity} A000045(n)/A000032(n). - Bruno Berselli, Jan 22 2018

From Christian Katzmann, Mar 19 2018: (Start)

Equals Sum_{n>=0} (2*n)!/(n!^2*3^(2*n+1)).

Equals Sum_{n>=0} 5*(2*n+1)!/(n!^2*3^(2*n+3)). (End)

Equals A010476/10. - R. J. Mathar, Jan 14 2021

Equals Sum_{k>=1} F(2^(k-1))/(L(2^k)+1) = Sum_{k>=0} A058635(k)/(A001566(k)+1), where F(k) = A000045(k) is the k-th Fibonacci number and L(k) = A000032(k) is the k-th Lucas number (Ohtsuka, 2014). - Amiram Eldar, Dec 09 2021

Extensions

More terms from Stefan Steinerberger, Apr 08 2006