A189961 - OEIS

%I #18 Mar 30 2024 15:56:50

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

%C 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 G. C. Greubel, <a href="/A189961/b189961.txt">Table of n, a(n) for n = 1..10000</a>

%F 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 [2,15,3,15,3,15,3,15,3,...]

%F From _Amiram Eldar_, Feb 06 2022: (Start)

%F Equals phi^4/sqrt(5) - 1, where phi is the golden ratio (A001622).

%F Equals lim_{k->oo} Fibonacci(k+4)/Lucas(k) - 1. (End)

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

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

%t FullSimplify[%]

%t N[%,130]

%t RealDigits[%]  (* A189961 *)

%t ContinuedFraction[%%]

%t RealDigits[(5+7*Sqrt[5])/10,10,150][[1]] (* _Harvey P. Dale_, Mar 30 2024 *)

%o (PARI) (5+7*sqrt(5))/10 \\ _G. C. Greubel_, Jan 13 2018

%o (Magma) (5+7*Sqrt(5))/10 // _G. C. Greubel_, Jan 13 2018

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

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, May 02 2011