cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A217684 Continued fraction expansion for log_10((1+sqrt(5))/2).

Original entry on oeis.org

0, 4, 1, 3, 1, 1, 1, 6, 4, 2, 1, 10, 1, 4, 46, 3, 1, 2, 1, 1, 1, 1, 3, 16, 2, 5, 1, 3, 2, 2, 9, 1, 1, 1, 2, 6, 106, 2, 3, 1, 3, 1, 1, 16, 20, 1, 1, 1, 4, 37, 1, 6, 1, 2, 6, 1, 1, 4, 2, 1, 2, 72, 10, 1, 1, 2, 3, 8, 1, 1, 1, 1, 1, 2, 1, 2, 3, 9, 1, 2, 4, 3, 2, 9, 1, 4, 2, 2, 2, 4
Offset: 0

Views

Author

V. Raman, Oct 11 2012

Keywords

Comments

The significance of this sequence is that the convergents of the continued fraction expansion of log_10((1+sqrt(5))/2) give the sequence of fractions p/q such that Lucas(q) gets increasingly closer to 10^p. For example, the first few convergents are 0/1, 1/4, 1/5, 4/19, 5/24, 9/43, 14/67, 93/445.
Clearly as we can see below
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. A097348 (decimal expansion), A217685/A217686 (convergents).

Programs

  • Mathematica
    ContinuedFraction[Log[10, GoldenRatio], 90] (* Jean-François Alcover, Oct 17 2012 *)
  • PARI
    default(realprecision, 99); contfrac(log((1+sqrt(5))/2)/log(10))

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

Original entry on oeis.org

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

Views

Author

V. Raman, Oct 11 2012

Keywords

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).

A217687 Values of n such that Fibonacci(n) gets increasingly closer to the powers of 10 (measured by the ratio between the Fibonacci number and the nearest power of 10).

Original entry on oeis.org

1, 2, 6, 11, 16, 83, 150, 217, 662, 1107, 2954, 346893, 690832, 1034771, 1378710, 1722649, 2066588, 2410527, 2754466, 3098405, 3442344, 3786283, 4130222, 4474161, 4818100, 5162039, 5505978, 5849917, 6193856, 6537795, 6881734, 7225673, 7569612, 7913551, 8257490, 8601429, 8945368, 9289307, 9633246, 9977185
Offset: 0

Views

Author

V. Raman, Oct 11 2012

Keywords

Comments

The sequence A217686 gives the sequence of values n such that Lucas(n) get increasingly closer to the powers of 10 (by the ratio between the Lucas number to the nearest power of 10).
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 A217686.

Crossrefs

Cf. A217684, A217685, A217686, A217688.

Programs

  • PARI
    default(realprecision, 1000); a=vector(100,i,(contfracpnqn(contfrac(log((1+sqrt(5))/2)/log(10), , 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])

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).

Original entry on oeis.org

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

Views

Author

V. Raman, Oct 11 2012

Keywords

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.

Crossrefs

Cf. A217684, A217685, A217686, A217687.

Programs

  • PARI
    default(realprecision, 1000); a=vector(100,i,(contfracpnqn(contfrac(log((1+sqrt(5))/2)/log(10), , 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])
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.