cp's OEIS Frontend

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

%I A217686 #17 Aug 07 2025 21:01:54
%S A217686 1,4,5,19,24,43,67,445,1847,4139,5986,63999,69985,343939,15891179,
%T A217686 48017476,63908655,175834786,239743441,415578227,655321668,1070899895,
%U A217686 3868021353,62959241543,129786504439,711891763738,841678268177,3236926568269,7315531404715,17867989377699
%N A217686 Denominators of the continued fraction convergents of log_10((1+sqrt(5))/2).
%C A217686 Lucas(Denominator of convergents) get increasingly closer to the values of 10^(Numerator of convergents).
%C A217686 For example,
%C A217686 Lucas(19) = 9349 ~ 10^4, error = 6.51%
%C A217686 Lucas(24) = 103682 ~ 10^5, error = 3.682%
%C A217686 Lucas(43) = 969323029 ~ 10^9, error = 3.068%
%C A217686 Lucas(67) = 100501350283429 ~ 10^14, error = 0.501%
%C A217686 In fact, for sufficiently large values of n, we will have that Lucas(n) ~ ((1+sqrt(5))/2)^n.
%F A217686 a(n) = A217684(n)*a(n-1) + a(n-2).
%o A217686 (PARI) default(realprecision, 21000); for(i=1, 100, print(contfracpnqn(contfrac(log((1+sqrt(5))/2)/log(10), , i))[2, 1]))
%Y A217686 Cf. A217684 (continued fraction expansion of log_10((1+sqrt(5))/2)).
%Y A217686 Cf. A217685 (numerators of the continued fraction convergents of log_10((1+sqrt(5))/2)).
%K A217686 nonn,cofr,frac
%O A217686 0,2
%A A217686 _V. Raman_, Oct 11 2012