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

0,1

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

Links

Table of n, a(n) for n=0..62.

Formula

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A001622, A010186.

Sequence in context: A091550 A027553 A159378 * A171223 A278896 A249397

Adjacent sequences: A172071 A172072 A172073 * A172075 A172076 A172077

Keyword

cofr,nonn,nice

Author

Shane Findley, Jan 25 2010

Extensions

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

Status

approved