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A217685
Numerators of the continued fraction convergents of log_10((1+sqrt(5))/2).
4
0, 1, 1, 4, 5, 9, 14, 93, 386, 865, 1251, 13375, 14626, 71879, 3321060, 10035059, 13356119, 36747297, 50103416, 86850713, 136954129, 223804842, 808368655, 13157703322, 27123775299, 148776579817, 175900355116, 676477645165, 1528855645446, 3734188936057
OFFSET
0,4
Lucas(Denominator of convergents) get increasingly closer to the values of 10^(Numerator of convergents).
For example,
Lucas(19) = 9349 ~ 10^4, error = 6.51%
Lucas(24) = 103682 ~ 10^5, error = 3.682%
Lucas(43) = 969323029 ~ 10^9, error = 3.068%
Lucas(67) = 100501350283429 ~ 10^14, error = 0.501%
In fact, for sufficiently large values of n, we will have that Lucas(n) ~ ((1+sqrt(5))/2)^n.
FORMULA
a(n) = A217684(n)*a(n-1) + a(n-2).
PROG
(PARI) default(realprecision, 21000); for(i=1, 100, print(contfracpnqn(contfrac(log((1+sqrt(5))/2)/log(10), 0, i))[1, 1]))
CROSSREFS
Cf. A217684 (continued fraction expansion of log_10((1+sqrt(5))/2)).
Cf. A217686 (denominators of the continued fraction convergents of log_10((1+sqrt(5))/2)).
Sequence in context: A121052 A041823 A042489 * A251632 A350695 A049860
KEYWORD
nonn,cofr,frac
AUTHOR
V. Raman, Oct 11 2012
STATUS
approved

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Last modified September 20 06:55 EDT 2024. Contains 376067 sequences. (Running on oeis4.)