A130234 Minimal index k of a Fibonacci number such that Fibonacci(k) >= n (the 'upper' Fibonacci Inverse).

0, 1, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 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, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset

0,3

Comments

Inverse of the Fibonacci sequence (A000045), nearly, since a(Fibonacci(n)) = n except for n = 2 (see A130233 for another version). a(n+1) is equal to the partial sum of the Fibonacci indicator sequence (see A104162).

Links

Table of n, a(n) for n=0..80.

Formula

a(n) = ceiling(log_phi((sqrt(5)*n + sqrt(5*n^2-4))/2)) = ceiling(arccosh(sqrt(5)*n/2)/log(phi)) where phi = (1+sqrt(5))/2, the golden ratio, for n > 0.

a(n) = A130233(n-1) + 1 for n > 0.

G.f.: x/(1-x) * Sum_{k >= 0} x^Fibonacci(k).

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

a(n) = A108852(n-1). - R. J. Mathar, Jan 31 2015

Example

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

Maple

A130234 := proc(n)
    local i;
    for i from 0 do
        if A000045(i) >= n then
            return i;
        end if;
    end do:
end proc: # R. J. Mathar, Jan 31 2015

Mathematica

a[n_] := For[i = 0, True, i++, If[Fibonacci[i] >= n, Return[i]]];
a /@ Range[0, 80] (* Jean-François Alcover, Apr 13 2020 *)

Prog

(PARI) a(n)=my(k); while(fibonacci(k)<n, k++); k \\ Charles R Greathouse IV, Feb 03 2014, corrected by M. F. Hasler, Apr 07 2021

Crossrefs

Partial sums: A130236.

Other related sequences: A000045, A130233, A130256, A130260, A104162, A108852.

Lucas inverse: A130241 - A130248.

Sequence in context: A370302 A098200 A092405 * A108852 A179413 A119476

Adjacent sequences: A130231 A130232 A130233 * A130235 A130236 A130237

Keyword

nonn,easy

Author

Hieronymus Fischer, May 17 2007

Status

approved