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A130260 Minimal index k of an even Fibonacci number A001906 such that A001906(k) = Fib(2k) >= n (the 'upper' even Fibonacci Inverse). 7
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 (list; graph; refs; listen; history; text; internal format)
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

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 A062537

Adjacent sequences:  A130257 A130258 A130259 * A130261 A130262 A130263

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified October 17 14:25 EDT 2018. Contains 316281 sequences. (Running on oeis4.)