A217685 - OEIS

%I #14 Aug 11 2024 09:55:24

%S 0,1,1,4,5,9,14,93,386,865,1251,13375,14626,71879,3321060,10035059,

%T 13356119,36747297,50103416,86850713,136954129,223804842,808368655,

%U 13157703322,27123775299,148776579817,175900355116,676477645165,1528855645446,3734188936057

%N Numerators 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 a(n) = A217684(n)*a(n-1) + a(n-2).

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

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

%Y Cf. A217686 (denominators of the continued fraction convergents of log_10((1+sqrt(5))/2)).

%K nonn,cofr,frac

%O 0,4

%A _V. Raman_, Oct 11 2012