A172074 Continued fraction expansion of sqrt(12500)+50 = 100*phi, where phi=(sqrt(5)+1)/2 is the golden ratio.

161, 1, 4, 11, 1, 1, 3, 6, 1, 13, 8, 1, 6, 1, 4, 1, 1, 2, 1, 1, 1, 1, 13, 2, 1, 3, 8, 1, 2, 19, 1, 54, 1, 19, 2, 1, 8, 3, 1, 2, 13, 1, 1, 1, 1, 2, 1, 1, 4, 1, 6, 1, 8, 13, 1, 6, 3, 1, 1, 11, 4, 1, 222

Offset: 0

Author

Shane Findley, Jan 25 2010

Keywords

• cofr
• nonn
• nice
• easy

Comments

The 62 trailing terms are repeated infinitely.

This is just one of an infinite set of continued fractions, related to the golden ratio, and more specifically to the square root of 125, 12500, 1250000...

Taking phi*10^k, one can look at sqrt(125) + 5, sqrt(12500) + 50 (this sequence), sqrt(1250000) + 500, etc.

This is not an efficient way to calculate phi. - Franklin T. Adams-Watters, Sep 10 2011

Periodic with a period of length 62, starting right after the initial term. Moreover, the sequence is symmetric when any 54 or 222 is taken as central value (cf. formula). - M. F. Hasler, Sep 09 2011

Crossrefs

Cf. A001622, A010186.

Programs

Mathematica

ContinuedFraction[N[Sqrt[12500], 50000], 63]
ContinuedFraction[100*GoldenRatio,100] (* Harvey P. Dale, Dec 30 2018 *)

PARI

default(realprecision, 199); contfrac((sqrt(5)+1)/.02)  \\ M. F. Hasler, Sep 09 2011

PARI

a(n)=[222-61*!n, 1, 4, 11, 1, 1, 3, 6, 1, 13, 8, 1, 6, 1, 4, 1, 1, 2, 1, 1, 1, 1, 13, 2, 1, 3, 8, 1, 2, 19, 1, 54][32-abs(n%62-31)]  \\ M. F. Hasler, Sep 09 2011

Formula

a(31*k - n) = a(31*k + n), for all n < 31k, k > 0. - M. F. Hasler, Sep 09 2011

Extensions

Clarified the definition, following an observation by Franklin T. Adams-Watters. M. F. Hasler, Sep 09 2011