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Decimal expansion of (5+9*sqrt(5))/12.
3

%I #9 Sep 08 2022 08:45:56

%S 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,

%T 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,

%U 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

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

%C 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.

%H G. C. Greubel, <a href="/A189963/b189963.txt">Table of n, a(n) for n = 1..10000</a>

%F 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:

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

%e 2.09371764979150893897354691821512384324713043637531095986983...

%t r=(1+5^(1/2))/2;

%t FromContinuedFraction[{r,r,r,r,r}]

%t FullSimplify[%]

%t N[%,130]

%t RealDigits[%] (*A189963*)

%t ContinuedFraction[%%]

%o (PARI) (5+9*sqrt(5))/12 \\ _G. C. Greubel_, Jan 13 2018

%o (Magma) (5+9*Sqrt(5))/12 // _G. C. Greubel_, Jan 13 2018

%Y Cf. A188635, A189961, A189962.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, May 02 2011