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A089959
a(n) = floor(1/(f(n) - f(n)^2)) with f(n) = frac(n*(sqrt(5) - 1)/2) (fractional part).
3
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).
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
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