A190180 Continued fraction of (1+sqrt(-3+4*sqrt(2)))/2.

1, 3, 5, 1, 2, 1, 1, 1, 2, 1, 12, 1, 5, 1, 1, 2, 1, 14, 2, 9, 11, 1, 12, 1, 2, 1, 832, 1, 2, 2, 5, 1, 1, 17, 1, 2, 1, 9, 1, 12, 1, 1, 1, 6, 3, 2, 1, 1, 6, 3, 1, 1, 1, 2, 2, 1, 3, 1, 3, 3, 1, 2, 1, 45, 1, 1, 1, 1, 62, 9, 1, 1, 2, 3, 1, 6, 1, 3, 5, 1

Offset: 1

Author

Clark Kimberling, May 05 2011

Keywords

• nonn
• cofr

Comments

Equivalent to the periodic continued fraction [1,r,1,r,...] where r=1+sqrt(2), the silver ratio. For geometric interpretations of both continued fractions, see A189979 and A188635.

1 followed by A190178.

Links

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

Crossrefs

Cf. A190179, A188635, A190177, A190178.

Programs

Magma

ContinuedFraction((1+Sqrt(-3+4*Sqrt(2)))/2); // G. C. Greubel, Dec 28 2017

Mathematica

r = 1 + 2^(1/2);
FromContinuedFraction[{1, r, {1, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190180 *)
RealDigits[N[%%, 120]]     (* A190179 *)
N[%%%, 40]
ContinuedFraction[(1 + Sqrt[-3 + 4*Sqrt[2]])/2, 100] (* G. C. Greubel, Dec 28 2017 *)

PARI

contfrac((1+sqrt(-3+4*sqrt(2)))/2) \\ G. C. Greubel, Dec 28 2017