A090038 a(n) = floor(1/({n*k}-{n*k}^2)) -1, where k = sqrt(2)-1 and {} is the fractional part.

3, 6, 4, 3, 14, 3, 10, 3, 4, 7, 3, 34, 3, 5, 4, 3, 24, 3, 7, 3, 3, 9, 3, 17, 3, 4, 5, 3, 82, 3, 6, 4, 3, 12, 3, 11, 3, 4, 6, 3, 58, 3, 5, 4, 3, 18, 3, 8, 3, 3, 8, 3, 21, 3, 4, 5, 3, 41, 3, 6, 4, 3, 10, 3, 13, 3, 4, 6, 3, 198, 3, 5, 4, 3, 15

Offset

1,1

Comments

a(P(n)) = A002203(n), n>=2, where P=A000129 are the Pell numbers.

Example: a(29) = 82, where 29 = P(5) and 82 = A002203(5).

a(A002203(n)) = P(n), n>=3.

Example: a(34) = 12, where 34 = A002203(4) and 12 = P(4).

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Example

a(9) = 4. Take {n*k} with k = .414213...= sqrt(2) - 1. Then {9*.414...} = .727922...with (.727922...)*(1 - .727922...) = .1980515...Invert, taking floor = 5. Finally, subtract 1 = 4.

Maple

A090038 := proc(n)
    k := sqrt(2)-1 ;
    fn := fpart(n*k) ;
    1/fn/(1-fn);
    floor(%)-1 ;
end proc: # R. J. Mathar, May 11 2013

Mathematica

Table[Floor[1/(FractionalPart[n*(Sqrt[2] - 1)] - FractionalPart[n*(Sqrt[2] - 1)]^2)] - 1, {n, 1, 100}] (* G. C. Greubel, Sep 27 2018 *)

Prog

(PARI) a(n) = floor(1/(frac(n*sqrt(2))-frac(n*sqrt(2))^2)) - 1; \\ Michel Marcus, Sep 28 2018

Crossrefs

Cf. A090038, A002203, A089959, A089960, A089961.

Sequence in context: A140072 A187148 A105559 * A308291 A006464 A233825

Adjacent sequences: A090035 A090036 A090037 * A090039 A090040 A090041

Keyword

nonn

Author

Gary W. Adamson, Nov 20 2003

Status

approved