A130233 a(n) is the maximal k such that Fibonacci(k) <= n (the "lower" Fibonacci Inverse).

0, 2, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset

0,2

Comments

Inverse of the Fibonacci sequence (A000045), nearly, since a(Fibonacci(n)) = n except for n = 1 (see A130234 for another version). a(n) + 1 is equal to the partial sum of the Fibonacci indicator sequence (see A104162).

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Formula

a(n) = floor(log_phi((sqrt(5)*n + sqrt(5*n^2+4))/2)) where phi = (1+sqrt(5))/2 = A001622.

a(n) = floor(arcsinh(sqrt(5)*n/2) / log(phi)), with log(phi) = A002390.

a(n) = A130234(n+1) - 1.

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

a(n) = floor(log_phi(sqrt(5)*n+1)), n >= 0, where phi is the golden ratio. - Hieronymus Fischer, Jul 02 2007

Example

a(10) = 6, since Fibonacci(6) = 8 <= 10 but Fibonacci(7) = 13 > 10.

Mathematica

fibLLog[0] := 0; fibLLog[1] := 2; fibLLog[n_Integer] := fibLLog[n] = If[n < Fibonacci[fibLLog[n - 1] + 1], fibLLog[n - 1], fibLLog[n - 1] + 1]; Table[fibLLog[n], {n, 0, 88}] (* Alonso del Arte, Sep 01 2013 *)

Prog

(PARI) a(n)=log(sqrt(5)*n+1.5)\log((1+sqrt(5))/2) \\ Charles R Greathouse IV, Mar 21 2012

Crossrefs

Cf. A130235 (partial sums), A104162 (first differences).

Other related sequences: A000045, A130234, A130237, A130239, A130255, A130259, A108852. Lucas inverse: A130241.

Cf. A001622 (golden ratio), A002390 (its log).

Sequence in context: A199769 A030601 A049839 * A131234 A349983 A204924

Adjacent sequences: A130230 A130231 A130232 * A130234 A130235 A130236

Keyword

nonn,easy

Author

Hieronymus Fischer, May 17 2007

Status

approved