A073732 Decimal expansion of lim_{n -> infinity} n*phi - Sum_{k=1..n} F(k+1)/F(k), where phi is the golden ratio and F(k) denotes the k-th Fibonacci number.

3, 1, 8, 4, 5, 2, 9, 6, 4, 0, 7, 4, 5, 0, 1, 0, 8, 1, 2, 9, 2, 1, 7, 5, 7, 2, 1, 3, 2, 6, 2, 4, 7, 6, 3, 9, 9, 3, 6, 1, 8, 7, 8, 2, 2, 7, 3, 0, 7, 5, 8, 3, 5, 2, 0, 9, 9, 0, 6, 4, 2, 6, 5, 9, 8, 4, 3, 4, 6, 8, 7, 8, 2, 6, 2, 1, 9, 0, 3, 3, 1, 1, 9, 0, 4, 9, 6, 5, 4, 1, 9, 6, 4, 5, 8, 2, 9, 6, 8, 7, 7, 0, 4, 7

Offset

0,1

Links

Table of n, a(n) for n=0..103.

Formula

Equals (1/2) * lim_{n -> infinity} n*sqrt(5) - Sum_{k=1..n} F(2*k)/F(k)^2.

From Amiram Eldar, Oct 05 2020: (Start)

Equals Sum_{k>=1} (-1)^(k+1)/(phi^k * F(k)).

Equals sqrt(5) * Sum_{k>=1} (-1)^(k+1)/(phi^(2*k) - (-1)^k). (End)

Example

0.31845296407450108129217572132624763993618782273...

Mathematica

f[n_] := f[n] = n *GoldenRatio - Sum[Fibonacci[k + 1]/Fibonacci[k], {k, 1, n}] // RealDigits[#, 10, 104]& // First; f[n=100]; While[f[n] != f[n-100], n = n+100]; f[n] (* Jean-François Alcover, Feb 13 2013 *)

Crossrefs

Cf. A000045, A001622.

Sequence in context: A055249 A125172 A344381 * A021318 A068437 A016477

Adjacent sequences: A073729 A073730 A073731 * A073733 A073734 A073735

Keyword

cons,easy,nonn

Author

Benoit Cloitre, Sep 01 2002

Status

approved