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

%I A189961 #18 Mar 30 2024 15:56:50
%S A189961 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,
%T A189961 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,
%U A189961 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
%N A189961 Decimal expansion of (5+7*sqrt(5))/10.
%C A189961 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.
%H A189961 G. C. Greubel, <a href="/A189961/b189961.txt">Table of n, a(n) for n = 1..10000</a>
%F A189961 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:
%F A189961 [2,15,3,15,3,15,3,15,3,...]
%F A189961 From _Amiram Eldar_, Feb 06 2022: (Start)
%F A189961 Equals phi^4/sqrt(5) - 1, where phi is the golden ratio (A001622).
%F A189961 Equals lim_{k->oo} Fibonacci(k+4)/Lucas(k) - 1. (End)
%t A189961 r=(1+5^(1/2))/2;
%t A189961 FromContinuedFraction[{r,r,r}]
%t A189961 FullSimplify[%]
%t A189961 N[%,130]
%t A189961 RealDigits[%]  (* A189961 *)
%t A189961 ContinuedFraction[%%]
%t A189961 RealDigits[(5+7*Sqrt[5])/10,10,150][[1]] (* _Harvey P. Dale_, Mar 30 2024 *)
%o A189961 (PARI) (5+7*sqrt(5))/10 \\ _G. C. Greubel_, Jan 13 2018
%o A189961 (Magma) (5+7*Sqrt(5))/10 // _G. C. Greubel_, Jan 13 2018
%Y A189961 Cf. A000032, A000045, A001622, A188635, A189962, A189963.
%K A189961 nonn,cons
%O A189961 1,1
%A A189961 _Clark Kimberling_, May 02 2011