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A217685 Numerators of the continued fraction convergents of log_10((1+sqrt(5))/2).

%I A217685 #22 Aug 08 2025 09:08:30
%S A217685 0,1,1,4,5,9,14,93,386,865,1251,13375,14626,71879,3321060,10035059,
%T A217685 13356119,36747297,50103416,86850713,136954129,223804842,808368655,
%U A217685 13157703322,27123775299,148776579817,175900355116,676477645165,1528855645446,3734188936057
%N A217685 Numerators of the continued fraction convergents of log_10((1+sqrt(5))/2).
%C A217685 Lucas(Denominator of convergents) get increasingly closer to the values of 10^(Numerator of convergents).
%C A217685 For example,
%C A217685   Lucas(19) = 9349 ~ 10^4, error = 6.51%
%C A217685   Lucas(24) = 103682 ~ 10^5, error = 3.682%
%C A217685   Lucas(43) = 969323029 ~ 10^9, error = 3.068%
%C A217685   Lucas(67) = 100501350283429 ~ 10^14, error = 0.501%
%C A217685 In fact, for sufficiently large values of n, we will have that Lucas(n) ~ ((1+sqrt(5))/2)^n.
%F A217685 a(n) = A217684(n)*a(n-1) + a(n-2).
%o A217685 (PARI) default(realprecision, 21000); a(n) = contfracpnqn(contfrac(log((1+sqrt(5))/2)/log(10), , n))[1, 1];
%Y A217685 Cf. A217684 (continued fraction expansion), A217686 (denominators).
%K A217685 nonn,cofr,frac
%O A217685 0,4
%A A217685 _V. Raman_, Oct 11 2012