A190183 Continued fraction of (1+x+sqrt(8+2x))/4, where x=sqrt(15).

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, 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, 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

Offset: 1

Author

Clark Kimberling, May 05 2011

Keywords

• nonn
• cofr

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A190182, A188635.

Programs

Magma

ContinuedFraction((1+Sqrt(15)+Sqrt(8+2*Sqrt(15)))/4); // G. C. Greubel, Dec 28 2017

Mathematica

r = (1 + 5^(1/2))/2;
FromContinuedFraction[{r, 1, 1, {r, 1, 1}}]
FullSimplify[%]
ContinuedFraction[%, 100] (* A190183 *)
RealDigits[N[%%, 120]]    (* A190182 *)
N[%%%, 40]
ContinuedFraction[(1+Sqrt[15]+Sqrt[8+2Sqrt[15]])/4,100] (* Harvey P. Dale, Apr 29 2013 *)

PARI

contfrac((1+sqrt(15)+sqrt(8+2*sqrt(15)))/4) \\ G. C. Greubel, Dec 28 2017