A190183 - OEIS

%I #12 Sep 08 2022 08:45:56

%S 2,4,1,3,10,1,3,1,1,2,66,1,4,2,1,1,48,5,1,1,2,1,1,1,8,2,1,1,4,16,2,2,

%T 1,4,1,3,1,3,1,11,1,1,8,16,1,1,1,10,1,2,4,1,1,1,3,1,1,1,1,30,1,1,2,1,

%U 1,8,13,1,1,6,6,1,6,1,1,2,2,10,1,2,7,9,2,4,7,3,1,2,2,1,2,5,4,2,3,2,3,2,1,3

%N Continued fraction of (1+x+sqrt(8+2x))/4, where x=sqrt(15).

%C Equivalent to the periodic continued fraction [r,1,1,r,1,1,...] where r=(1+sqrt(5))/2, the golden ratio. For geometric interpretations of both continued fractions, see A190182 and A188635.

%H G. C. Greubel, <a href="/A190183/b190183.txt">Table of n, a(n) for n = 1..10000</a>

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

%t FromContinuedFraction[{r, 1, 1, {r, 1, 1}}]

%t FullSimplify[%]

%t ContinuedFraction[%, 100] (* A190183 *)

%t RealDigits[N[%%, 120]]    (* A190182 *)

%t N[%%%, 40]

%t ContinuedFraction[(1+Sqrt[15]+Sqrt[8+2Sqrt[15]])/4,100] (* _Harvey P. Dale_, Apr 29 2013 *)

%o (PARI) contfrac((1+sqrt(15)+sqrt(8+2*sqrt(15)))/4) \\ _G. C. Greubel_, Dec 28 2017

%o (Magma) ContinuedFraction((1+Sqrt(15)+Sqrt(8+2*Sqrt(15)))/4); // _G. C. Greubel_, Dec 28 2017

%Y Cf. A190182, A188635.

%K nonn,cofr

%O 1,1

%A _Clark Kimberling_, May 05 2011