OFFSET
0,2
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.
FORMULA
MATHEMATICA
ContinuedFraction[Log[10, GoldenRatio], 90] (* Jean-François Alcover, Oct 17 2012 *)
PROG
(PARI) default(realprecision, 99); contfrac(log((1+sqrt(5))/2)/log(10))
CROSSREFS
KEYWORD
nonn,cofr
AUTHOR
V. Raman, Oct 11 2012
STATUS
approved