A125889 Denominator of sum of first n ratios of Fibonacci to Lucas numbers.

1, 1, 3, 6, 42, 462, 1386, 40194, 1889118, 17946621, 735811461, 146426480739, 3367809056997, 1754628518695437, 493050613753417797, 30569138052711903414, 67466087682335170834698, 240921399113618895050706558, 77335769115471665311276805118

Offset

0,3

Comments

A125888(n)/a(n) = Sum_{i=1..n} A000045(i)/A000032(i) = n/sqrt(5) + O(1). - Max Alekseyev, Dec 07 2006

GCD(F(i),L(i)) <= 2, so the ratio reduces when there is a factor of two in common, every third term. Example as continued fraction: 0 + 1 + 1/3 + 2/4 + 3/7 + 5/11 + 8/18 + 13/29 + 21/47 + 34/76 + 55/123 = 4 + 1/(1 + 1/(19 + 1/(4 + 1/(1 + 1/(13 + 1/(2 + 1/(4 + 1/(1 + 1/(6 + 1/(6 + 1/(1 + 1/(85 + 1/(1 + 1/(4 + 1/2)))))))))))))).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Formula

a(n) = denominator(Sum_{i=1..n} F(i)/L(i));

a(n) = denominator(Sum_{i=1..n} A000045(i)/A000032(i)).

Example

The fractions, reduced to lowest terms, begin: 0/1, 1/1, 4/3, 11/6, 95/42, 1255/462, 4381/1386, 145067/40194, 7662223/1889118, 80819870/17946621, 3642636055/735811461, ...

Mathematica

With[{nn=20}, Join[{1}, Denominator[Accumulate[Fibonacci[Range[nn]]/ LucasL[ Range[ nn]]]]]] (* Harvey P. Dale, Mar 21 2015 *)

Prog

(Magma) [1] cat [Denominator(&+[Fibonacci(i)/Lucas(i): i in [1..n]]): n in [1..25]]; // Vincenzo Librandi, Mar 25 2017

Crossrefs

Cf. A000032, A000045, A125888.

Sequence in context: A109491 A085696 A079095 * A300599 A002028 A202858

Adjacent sequences: A125886 A125887 A125888 * A125890 A125891 A125892

Keyword

easy,frac,nonn

Author

Jonathan Vos Post, Dec 13 2006

Extensions

More terms and edited by Jon E. Schoenfield, Mar 16 2014

Status

approved