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A189961 Decimal expansion of (5+7*sqrt(5))/10. 3
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
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)
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 *)
PROG
(PARI) (5+7*sqrt(5))/10 \\ G. C. Greubel, Jan 13 2018
(Magma) (5+7*Sqrt(5))/10 // G. C. Greubel, Jan 13 2018
CROSSREFS
Sequence in context: A142457 A100711 A199464 * A211241 A140247 A271170
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, May 02 2011
STATUS
approved

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Last modified April 16 19:21 EDT 2024. Contains 371754 sequences. (Running on oeis4.)