A134972 Decimal expansion of 2 divided by golden ratio = 2/phi = 4/(1 + sqrt(5)) = 2*(-1 + phi).

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

Offset

1,2

Comments

Convergents are 4/2, 8/8, 32/24, 96/80, 320/256, 1024/832, 3328/2688, 10752/8704, 34816/28160, 112640/91136, 364544/294912, 1179648/954368, 3817472/3088384, 12353536/9994240, ... = A209084/A063727. - Seiichi Kirikami, Mar 14 2012

2*(-1 + phi) is an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Feb 16 2016

Links

Table of n, a(n) for n=1..104.

Michael Penn, How large is the blue ▲, YouTube video, 2021.

Formula

Equals A134945 - 2 = A002163 - 1 = A098317 - 3. - R. J. Mathar, Oct 27 2008

2*(-1 + A001622). - Wolfdieter Lang, Feb 17 2016

Equals the harmonic mean of 1 and phi, 2*phi/(1+phi). - Stanislav Sykora, Apr 11 2016

From Christian Katzmann, Mar 19 2018: (Start)

Equals Sum_{n>=0} (15*(2*n)!-8*n!^2)/(n!^2*3^(2*n+2)).

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

Equals 1/A019863. - R. J. Mathar, Jan 17 2021

Equals 2*sin(Pi/5)/sin(2*Pi/5) = hypergeom([1/5, 3/5], [7/5], 1) = hypergeom([-1/5, -3/5], [3/5], 1). - Peter Bala, Mar 04 2022

Example

1.236067977499789696...

Mathematica

RealDigits[ N[4/(1+Sqrt[5]), 150] ] [ [1] ] (* Seiichi Kirikami, Mar 14 2012 *)

Prog

(PARI) 4/(1+sqrt(5)) \\ Altug Alkan, Apr 11 2016

Crossrefs

Cf. A001622, A019863, A063727, A209084, A033887.

Sequence in context: A331205 A075174 A075176 * A078890 A021813 A082052

Adjacent sequences: A134969 A134970 A134971 * A134973 A134974 A134975

Keyword

nonn,easy,cons

Author

Omar E. Pol, Nov 15 2007

Status

approved