This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A217684 #34 Aug 08 2025 09:08:40 %S A217684 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 A217684 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 A217684 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 A217684 Continued fraction expansion for log_10((1+sqrt(5))/2). %C A217684 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 A217684 Clearly as we can see below %C A217684 Lucas(19) = 9349 ~ 10^4, error = 6.51% %C A217684 Lucas(24) = 103682 ~ 10^5, error = 3.682% %C A217684 Lucas(43) = 969323029 ~ 10^9, error = 3.068% %C A217684 Lucas(67) = 100501350283429 ~ 10^14, error = 0.501% %C A217684 In fact, for sufficiently large values of n, we will have that Lucas(n) ~ ((1+sqrt(5))/2)^n. %t A217684 ContinuedFraction[Log[10, GoldenRatio], 90] (* _Jean-François Alcover_, Oct 17 2012 *) %o A217684 (PARI) default(realprecision, 99); contfrac(log((1+sqrt(5))/2)/log(10)) %Y A217684 Cf. A097348 (decimal expansion), A217685/A217686 (convergents). %K A217684 nonn,cofr %O A217684 0,2 %A A217684 _V. Raman_, Oct 11 2012