A217688 Values of n such that 10^n gets increasingly closer to a Fibonacci number (measured by the ratio between the power of 10 and the nearest Fibonacci number).

0, 1, 2, 3, 17, 31, 45, 138, 231, 617, 72496, 144375, 216254, 288133, 360012, 431891, 503770, 575649, 647528, 719407, 791286, 863165, 935044, 1006923, 1078802, 1150681, 1222560, 1294439, 1366318, 1438197, 1510076, 1581955, 1653834, 1725713, 1797592, 1869471, 1941350, 2013229, 2085108, 2156987

Offset

1,3

Comments

The sequence A217685 gives the sequence of values n such that 10^n gets increasingly closer to a Lucas number.

Given that for sufficiently large values of n, Fibonacci(n) ~ Lucas(n)/sqrt(5) ~ (((1+sqrt(5))/2)^n)/(sqrt(5)), the intermediate differences between the terms in this sequence also need to be a member of the sequence A217685.

Links

Table of n, a(n) for n=1..40.

Prog

(PARI) default(realprecision, 1000); a=vector(100, i, (contfracpnqn(contfrac(log((1+sqrt(5))/2)/log(10), 0, i))[2, 1]))
log_fibonacci(j)=(j*log((1+sqrt(5))/2)/log(10))-(log(sqrt(5))/log(10))
deviation(k)=abs(round(log_fibonacci(k))-log_fibonacci(k))
n=6; err=deviation(n); m=3; while(n<10^20, if(deviation(n+a[m])<err, n=n+a[m]; err=deviation(n); print(round(log_fibonacci(n))), m++))

Crossrefs

Cf. A217684, A217685, A217686, A217687.

Sequence in context: A215280 A219559 A193051 * A263570 A029733 A153686

Adjacent sequences: A217685 A217686 A217687 * A217689 A217690 A217691

Keyword

nonn,base

Author

V. Raman, Oct 11 2012

Status

approved