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 A217686 Denominators of the continued fraction convergents of log_10((1+sqrt(5))/2). 5

%I

%S 1,4,5,19,24,43,67,445,1847,4139,5986,63999,69985,343939,15891179,

%T 48017476,63908655,175834786,239743441,415578227,655321668,1070899895,

%U 3868021353,62959241543,129786504439,711891763738,841678268177,3236926568269,7315531404715,17867989377699

%N Denominators of the continued fraction convergents of log_10((1+sqrt(5))/2).

%C Lucas(Denominator of convergents) get increasingly closer to the values of 10^(Numerator of convergents).

%C For example,

%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 A217686(n) = A217684(n)*A217686(n-1) + A217686(n-2).

%o (PARI) default(realprecision, 21000); for(i=1, 100, print(contfracpnqn(contfrac(log((1+sqrt(5))/2)/log(10), 0, i))[2, 1]))

%Y Cf. A217684 (continued fraction expansion of log_10((1+sqrt(5))/2)).

%Y Cf. A217685 (numerators of the continued fraction convergents of log_10((1+sqrt(5))/2)).

%K nonn,cofr

%O 0,2

%A _V. Raman_, Oct 11 2012

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Last modified October 28 13:26 EDT 2021. Contains 348329 sequences. (Running on oeis4.)