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

Author

Hieronymus Fischer, May 25 2007, Jul 02 2007

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

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

Programs

Magma

[Floor(Log((Sqrt(5)*n+1))/(2*Log((1+Sqrt(5))/2))): n in [0..100]]; // G. C. Greubel, Sep 12 2018

Mathematica

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

PARI

vector(100, n, n--; floor(log((sqrt(5)*n+1))/(2*log((1+sqrt(5))/2) ))) \\ G. C. Greubel, Sep 12 2018

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.