A189961 Decimal expansion of (5+7*sqrt(5))/10.

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

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 3 golden rectangles. For a general discussion, see A188635.

Links

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

Crossrefs

Cf. A000032, A000045, A001622, A188635, A189962, A189963.

Programs

Magma

(5+7*Sqrt(5))/10 // G. C. Greubel, Jan 13 2018

Mathematica

r=(1+5^(1/2))/2;
FromContinuedFraction[{r,r,r}]
FullSimplify[%]
N[%,130]
RealDigits[%]  (* A189961 *)
ContinuedFraction[%%]
RealDigits[(5+7*Sqrt[5])/10,10,150][[1]] (* Harvey P. Dale, Mar 30 2024 *)

PARI

(5+7*sqrt(5))/10 \\ G. C. Greubel, Jan 13 2018

Formula

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

[2,15,3,15,3,15,3,15,3,...]

From Amiram Eldar, Feb 06 2022: (Start)

Equals phi^4/sqrt(5) - 1, where phi is the golden ratio (A001622).

Equals lim_{k->oo} Fibonacci(k+4)/Lucas(k) - 1. (End)