A217684 Continued fraction expansion for log_10((1+sqrt(5))/2).

0, 4, 1, 3, 1, 1, 1, 6, 4, 2, 1, 10, 1, 4, 46, 3, 1, 2, 1, 1, 1, 1, 3, 16, 2, 5, 1, 3, 2, 2, 9, 1, 1, 1, 2, 6, 106, 2, 3, 1, 3, 1, 1, 16, 20, 1, 1, 1, 4, 37, 1, 6, 1, 2, 6, 1, 1, 4, 2, 1, 2, 72, 10, 1, 1, 2, 3, 8, 1, 1, 1, 1, 1, 2, 1, 2, 3, 9, 1, 2, 4, 3, 2, 9, 1, 4, 2, 2, 2, 4

Offset: 0

Author

V. Raman, Oct 11 2012

Keywords

• nonn
• cofr

Comments

The significance of this sequence is that the convergents of the continued fraction expansion of log_10((1+sqrt(5))/2) give the sequence of fractions p/q such that Lucas(q) gets increasingly closer to 10^p. For example, the first few convergents are 0/1, 1/4, 1/5, 4/19, 5/24, 9/43, 14/67, 93/445.

Clearly as we can see below

Lucas(19) = 9349 ~ 10^4, error = 6.51%

Lucas(24) = 103682 ~ 10^5, error = 3.682%

Lucas(43) = 969323029 ~ 10^9, error = 3.068%

Lucas(67) = 100501350283429 ~ 10^14, error = 0.501%

In fact, for sufficiently large values of n, we will have that Lucas(n) ~ ((1+sqrt(5))/2)^n.

Crossrefs

Cf. A097348 (decimal expansion), A217685/A217686 (convergents).

Programs

Mathematica

ContinuedFraction[Log[10, GoldenRatio], 90] (* Jean-François Alcover, Oct 17 2012 *)

PARI

default(realprecision, 99); contfrac(log((1+sqrt(5))/2)/log(10))