A189962 -id:A189962 - cp's OEIS frontend

A189963 Decimal expansion of (5+9*sqrt(5))/12.

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

Offset: 1

Clark Kimberling, May 02 2011

The constant at A189963 is the shape of a rectangle whose continued fraction partition consists of 5 golden rectangles. For a general discussion, see A188635.

			2.09371764979150893897354691821512384324713043637531095986983...

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

Cf. A188635, A189961, A189962.

(5+9*Sqrt(5))/12 // G. C. Greubel, Jan 13 2018

r=(1+5^(1/2))/2;
FromContinuedFraction[{r,r,r,r,r}]
FullSimplify[%]
N[%,130]
RealDigits[%]  (*A189963*)
ContinuedFraction[%%]

(5+9*sqrt(5))/12 \\ G. C. Greubel, Jan 13 2018

Continued fraction (as explained at A189959): [r,r,r,r,r], where r=(1+sqrt(5))/2. Ordinary continued fraction, as given by Mathematica program shown below:

[2,10,1,2,29,1,5,2,1,1,2,1,3,5,1,3,3,10,1,2,29,...].

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

Original entry on oeis.org

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

Clark Kimberling, May 02 2011

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.

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

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

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

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 *)

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

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)

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A189963 Decimal expansion of (5+9*sqrt(5))/12.

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