

A130234


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


23



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
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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^24))/2)) = ceiling(arccosh(sqrt(5)*n/2)/log(phi)) where phi = (1+sqrt(5))/2, the golden ratio, for n > 0.
a(n) = A130233(n1) + 1 for n > 0.
G.f.: x/(1x) * 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(n1).  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] (* JeanFrançois 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: A266898 A098200 A092405 * A108852 A179413 A119476
Adjacent sequences: A130231 A130232 A130233 * A130235 A130236 A130237


KEYWORD

nonn,easy


AUTHOR

Hieronymus Fischer, May 17 2007


STATUS

approved



