A217686 Denominators of the continued fraction convergents of log_10((1+sqrt(5))/2).

1, 4, 5, 19, 24, 43, 67, 445, 1847, 4139, 5986, 63999, 69985, 343939, 15891179, 48017476, 63908655, 175834786, 239743441, 415578227, 655321668, 1070899895, 3868021353, 62959241543, 129786504439, 711891763738, 841678268177, 3236926568269, 7315531404715, 17867989377699

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 of log_10((1+sqrt(5))/2)).

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

Programs

PARI

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

Formula

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