A130260 Minimal index k of an even Fibonacci number A001906 such that A001906(k) = Fib(2k) >= n (the 'upper' even Fibonacci Inverse).

0, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset

0,3

Comments

Inverse of the even Fibonacci sequence (A001906), since a(A001906(n))=n (see A130259 for another version).

a(n+1) is the number of even Fibonacci numbers (A001906) <=n.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36.

Formula

a(n) = ceiling(arcsinh(sqrt(5)*n/2)/(2*log(phi))) for n>=0.

a(n) = ceiling(arccosh(sqrt(5)*n/2)/(2*log(phi))) for n>=1.

a(n) = ceiling(log_phi(sqrt(5)*n)/2)=ceiling(log_phi(sqrt(5)*n-1)/2) for n>=1, where phi=(1+sqrt(5))/2.

a(n) = A130259(n-1) + 1, for n>=1.

G.f.: g(x)=x/(1-x)*Sum_{k>=0} x^Fib(2*k).

Example

a(10)=4 because A001906(4)=21>=10, but A001906(3)=8<10.

Mathematica

Join[{0}, Table[Ceiling[Log[GoldenRatio, Sqrt[5]*n]/2], {n, 1, 100}]] (* G. C. Greubel, Sep 12 2018 *)

Prog

(PARI) for(n=0, 100, print1(if(n==0, 0, ceil(log(sqrt(5)*n)/(2*log((1+ sqrt(5))/2)))), ", ")) \\ G. C. Greubel, Sep 12 2018
(Magma) [0] cat [Ceiling(Log(Sqrt(5)*n)/(2*Log((1+ Sqrt(5))/2))): n in [1..100]]; // G. C. Greubel, Sep 12 2018

Crossrefs

Cf. partial sums A130262. Other related sequences: A000045, A001519, A130234, A130237, A130239, A130256, A130259. Lucas inverse: A130241 - A130248.

Sequence in context: A161358 A120699 A072643 * A276621 A111393 A323665

Adjacent sequences: A130257 A130258 A130259 * A130261 A130262 A130263

Keyword

nonn

Author

Hieronymus Fischer, May 25 2007, May 28 2007, Jul 02 2007

Status

approved