A130255 Maximal index k of an odd Fibonacci number (A001519) such that A001519(k) = Fibonacci(2k-1) <= n (the 'lower' odd Fibonacci Inverse).

1, 2, 2, 2, 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, 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, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Author

Hieronymus Fischer, May 24 2007, Jul 02 2007

Keywords

• nonn

Comments

Inverse of the odd Fibonacci sequence (A001519), nearly, since a(A001519(n))=n except for n=0 (see A130256 for another version). a(n)+1 is the number of odd Fibonacci numbers (A001519) <= n (for n >= 1).

Examples

			a(10)=3 because A001519(3) = 5 <= 10, but A001519(4) = 13 > 10.
		

Links

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

Crossrefs

Cf. partial sums A130257. Other related sequences: A000045, A130233, A130237, A130239, A130256, A130259, A104160. Lucas inverse: A130241 - A130248.

Programs

Magma

phi:=(1+Sqrt(5))/2; [Floor((1 +Argsinh(Sqrt(5)*n/2)/Log(phi))/2): n in [1..100]]; // G. C. Greubel, Sep 09 2018

Mathematica

Table[Floor[(1 +ArcSinh[Sqrt[5]*n/2]/Log[GoldenRatio])/2], {n, 1, 100}] (* G. C. Greubel, Sep 09 2018 *)

PARI

phi=(1+sqrt(5))/2; vector(100, n, floor((1 +asinh(sqrt(5)*n/2)/log(phi))/2)) \\ G. C. Greubel, Sep 09 2018

Formula

a(n) = floor((1 + arcsinh(sqrt(5)*n/2)/log(phi))/2).

a(n) = floor((1 + arccosh(sqrt(5)*n/2)/log(phi))/2).

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

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

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