A190185 Continued fraction of sqrt(1+x+sqrt(1+2*x)), where x=sqrt(2/3).

1, 1, 5, 1, 6, 1, 5, 1, 1, 40, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 5, 1, 15, 1, 3, 1, 2, 2, 5, 1, 1, 1, 1, 4, 5, 65, 1, 13, 1, 3, 4, 1, 1, 1, 4, 13, 1, 1, 2, 1, 3, 2, 2, 1, 10, 1, 20, 4, 15, 6, 1, 3, 10, 1, 78, 1, 1, 11, 15, 1, 11, 179, 2, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 3, 2, 6, 1, 1, 7, 5, 1, 4, 1, 9, 1, 1, 2, 10, 3

Offset: 1

Author

Clark Kimberling, May 05 2011

Keywords

• nonn
• cofr

Comments

Equivalent to the periodic continued fraction [sqrt(2), sqrt(3), sqrt(2), sqrt(3),...]. For geometric interpretations of both continued fractions, see A190184 and A188635.

Links

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

Crossrefs

Cf. A188635, A190184.

Programs

Magma

ContinuedFraction(Sqrt(1 + Sqrt(2/3) + Sqrt(1 + 2*Sqrt(2/3)))); // G. C. Greubel, Dec 28 2017

Mathematica

FromContinuedFraction[{2^(1/2), 3^(1/2), {2^(1/2), 3^(1/2)}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190185 *)
RealDigits[N[%%, 120]]      (* A190186 *)
N[%%%, 40]
ContinuedFraction[Sqrt[1 + Sqrt[2/3] + Sqrt[1 + 2*Sqrt[2/3]]], 100] (* G. C. Greubel, Dec 28 2017 *)

PARI

contfrac(sqrt(1 + sqrt(2/3) + sqrt(1 + 2*sqrt(2/3)))) \\ G. C. Greubel, Dec 28 2017

Extensions

Definition corrected by Bruno Berselli, May 13 2011