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%I #26 Oct 18 2012 04:14:30
%S 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,
%T 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,
%U 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
%N Continued fraction expansion for log_10((1+sqrt(5))/2).
%C 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.
%C Clearly as we can see below
%C Lucas(19) = 9349 ~ 10^4, error = 6.51%
%C Lucas(24) = 103682 ~ 10^5, error = 3.682%
%C Lucas(43) = 969323029 ~ 10^9, error = 3.068%
%C Lucas(67) = 100501350283429 ~ 10^14, error = 0.501%
%C In fact, for sufficiently large values of n, we will have that Lucas(n) ~ ((1+sqrt(5))/2)^n.
%F A217685(n) = a(n)*A217685(n-1) + A217685(n-2).
%F A217686(n) = a(n)*A217686(n-1) + A217686(n-2).
%t ContinuedFraction[Log[10, GoldenRatio], 90] (* _Jean-François Alcover_, Oct 17 2012 *)
%o (PARI) default(realprecision, 99); contfrac(log((1+sqrt(5))/2)/log(10))
%Y Cf. A097348, A217685, A217686.
%K nonn,cofr
%O 0,2
%A _V. Raman_, Oct 11 2012
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