

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
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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,...]
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]
ContinuedFraction[%%]
RealDigits[(5+7*Sqrt[5])/10, 10, 150][[1]] (* Harvey P. Dale, Mar 30 2024 *)


PROG



CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



