A130259 Maximal index k of an even Fibonacci number (A001906) such that A001906(k) = Fib(2k) <= n (the 'lower' even Fibonacci Inverse).

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

Offset

0,4

Comments

Inverse of the even Fibonacci sequence (A001906), since a(A001906(n))=n (see A130260 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) = floor(arcsinh(sqrt(5)*n/2)/(2*log(phi))), where phi=(1+sqrt(5))/2.

a(n) = A130260(n+1) - 1.

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

a(n) = floor(1/2*log_phi(sqrt(5)*n+1)) for n>=0.

Example

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

Mathematica

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

Prog

(PARI) vector(100, n, n--; floor(log((sqrt(5)*n+1))/(2*log((1+sqrt(5))/2) ))) \\ G. C. Greubel, Sep 12 2018
(Magma) [Floor(Log((Sqrt(5)*n+1))/(2*Log((1+Sqrt(5))/2))): n in [0..100]]; // G. C. Greubel, Sep 12 2018

Crossrefs

Cf. partial sums A130261. Other related sequences: A000045, A001519, A130233, A130237, A130239, A130255, A130260, A104160. Lucas inverse: A130241 - A130248.

Sequence in context: A105513 A004233 A363832 * A068549 A132173 A023968

Adjacent sequences: A130256 A130257 A130258 * A130260 A130261 A130262

Keyword

nonn

Author

Hieronymus Fischer, May 25 2007, Jul 02 2007

Status

approved