A185048 Length of the continued fraction for floor(Fibonacci(n)*(1+sqrt(5))/2) / Fibonacci(n).

1, 1, 2, 2, 4, 2, 6, 4, 8, 4, 10, 6, 12, 6, 14, 8, 16, 8, 18, 10, 20, 10, 22, 12, 24, 12, 26, 14, 28, 14, 30, 16, 32, 16, 34, 18, 36, 18, 38, 20, 40, 20, 42, 22, 44, 22, 46, 24, 48, 24, 50, 26, 52, 26, 54, 28, 56, 28, 58, 30, 60, 30, 62, 32, 64, 32, 66, 34, 68, 34

Offset: 1

Author

Benoit Cloitre, Feb 15 2011

Keywords

• nonn
• easy

Comments

A240828 is an essentially identical sequence. - Bruno Berselli, Apr 18 2014

Links

• Bruno Berselli, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000045, A001622, A240828.

Programs

Mathematica

Table[Length[ContinuedFraction[Floor[Fibonacci[n]*GoldenRatio]/Fibonacci[n]]], {n, 70}]

PARI

a(n)=if(n<3,1,if(n%2,n-1,2*floor(n/4)))

Formula

a(1)=a(2)=1, for k>=2 we have a(2k)=2*floor(k/2) and a(2k-1)=2*k-2.

G.f.: x*(x^7+x^6-x^5+x^4+x^3+x^2+x+1) / ((x-1)^2*(x+1)^2*(x^2+1)). - Colin Barker, Jun 20 2013