|
|
A217685
|
|
Numerators of the continued fraction convergents of log_10((1+sqrt(5))/2).
|
|
5
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
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.
|
|
LINKS
|
|
|
FORMULA
|
|
|
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)).
|
|
KEYWORD
|
nonn,cofr
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|