A189962 Decimal expansion of 3*(1 + 3*sqrt(5))/11.

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

Offset: 1

Author

Clark Kimberling, May 02 2011

Keywords

• nonn
• cons

Comments

The constant at A189961 is the shape of a rectangle whose continued fraction partition consists of 4 golden rectangles. For a general discussion, see A188635.

Examples

			2.10223743613619156978932391078013510172414229422761...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A188635, A189961, A189963.

Programs

Magma

3*(1+3*Sqrt(5))/11 // G. C. Greubel, Jan 13 2018

Mathematica

r=(1+5^(1/2))/2;
FromContinuedFraction[{r,r,r,r}]
FullSimplify[%]
N[%,130]
RealDigits[%]  (*A189962*)
ContinuedFraction[%%]
RealDigits[3 (1+3*Sqrt[5])/11,10,150][[1]] (* Harvey P. Dale, Sep 11 2023 *)

PARI

3*(1+3*sqrt(5))/11 \\ G. C. Greubel, Jan 13 2018

Formula

Continued fraction (as explained at A188635): [r,r,r,r], where r = (1 + sqrt(5))/2. Ordinary continued fraction, as given by Mathematica program shown below:

[2,9,1,3,1,1,3,9,1,3,1,1,3,9,1,3,1,1,3,...]

Extensions

Definition corrected by G. C. Greubel, Jan 13 2018