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

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

Author

V. Raman, Oct 11 2012

Keywords

• nonn
• cofr
• frac

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.

Crossrefs

Cf. A217684 (continued fraction expansion), A217686 (denominators).

Programs

PARI

default(realprecision, 21000); a(n) = contfracpnqn(contfrac(log((1+sqrt(5))/2)/log(10), , n))[1, 1];

Formula

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