A089959 a(n) = floor(1/(f(n) - f(n)^2)) with f(n) = frac(n*(sqrt(5) - 1)/2) (fractional part).

4, 5, 8, 4, 12, 4, 4, 19, 4, 6, 6, 4, 30, 4, 5, 10, 4, 9, 5, 4, 48, 4, 5, 7, 4, 15, 4, 4, 14, 4, 7, 5, 4, 77, 4, 5, 8, 4, 10, 4, 4, 24, 4, 6, 6, 4, 22, 4, 4, 11, 4, 8, 5, 4, 124, 4, 5, 7, 4, 13, 4, 4, 16, 4, 7, 6, 4, 39, 4, 5, 9, 4, 9, 5, 4, 35, 4, 6, 6, 4

Offset

1,1

Comments

Denote by Fn and Ln the Fibonacci resp. Lucas numbers. Then some of the terms follow one of the following two patterns: (1) a(Fn) = (Ln + 1). Example: a(8) = 19 since 8 = F6 and 18 = L6. (2) a(Ln) = (Fn + 1). Example: a(29) = 14 = (F7 + 1) = (13 + 1).

Links

Table of n, a(n) for n=1..80.

Formula

a(n) = floor(1/(n*k*(1 - n*k)); k = phi^(-1).

Example

a(7) = 4 = floor( 1/(.3262379...)*(.67376207...); where {x} = fractional part of x = (7)*(.6180339...)= .3262379...; (1 - {x}) = .67376207...; .6180339... = (sqrt(5)-1)/2 = phi^(-1).

Mathematica

Table[Floor[1/(FractionalPart[(2*n)/(1+Sqrt[5])]*(1-FractionalPart[ (2*n)/(1 + Sqrt[5])]))], {n, 1, 80}] (* Stefan Steinerberger, Jul 01 2007 *)

Prog

(PARI) default(realprecision, 200); p=(sqrt(5)-1)/2; vector(100, n, 1\(frac(n*p)-frac(n*p)^2)) \\ M. F. Hasler, Apr 06 2009

Crossrefs

Sequence in context: A143717 A155921 A016721 * A085996 A296495 A020804

Adjacent sequences: A089956 A089957 A089958 * A089960 A089961 A089962

Keyword

nonn

Author

Gary W. Adamson, Nov 16 2003

Extensions

Corrected and extended by Stefan Steinerberger, Jul 01 2007

Definition, comment and example reworded and corrected by M. F. Hasler, Apr 06 2009

Status

approved