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A217684
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Continued fraction expansion for log_10((1+sqrt(5))/2).
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4
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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MATHEMATICA
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PROG
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(PARI) default(realprecision, 99); contfrac(log((1+sqrt(5))/2)/log(10))
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CROSSREFS
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KEYWORD
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nonn,cofr
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AUTHOR
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STATUS
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approved
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