cp's OEIS Frontend

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

%I A130234 #37 Apr 10 2021 22:38:41
%S A130234 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,
%T A130234 9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,
%U A130234 11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11
%N A130234 Minimal index k of a Fibonacci number such that Fibonacci(k) >= n (the 'upper' Fibonacci Inverse).
%C A130234 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).
%F A130234 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.
%F A130234 a(n) = A130233(n-1) + 1 for n > 0.
%F A130234 G.f.: x/(1-x) * Sum_{k >= 0} x^Fibonacci(k).
%F A130234 a(n) = ceiling(log_phi(sqrt(5)*n - 1)) for n > 0, where phi is the golden ratio. - _Hieronymus Fischer_, Jul 02 2007
%F A130234 a(n) = A108852(n-1). - _R. J. Mathar_, Jan 31 2015
%e A130234 a(10) = 7, since Fibonacci(7) = 13 >= 10 but Fibonacci(6) = 8 < 10.
%p A130234 A130234 := proc(n)
%p A130234     local i;
%p A130234     for i from 0 do
%p A130234         if A000045(i) >= n then
%p A130234             return i;
%p A130234         end if;
%p A130234     end do:
%p A130234 end proc: # _R. J. Mathar_, Jan 31 2015
%t A130234 a[n_] := For[i = 0, True, i++, If[Fibonacci[i] >= n, Return[i]]];
%t A130234 a /@ Range[0, 80] (* _Jean-François Alcover_, Apr 13 2020 *)
%o A130234 (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
%Y A130234 Partial sums: A130236.
%Y A130234 Other related sequences: A000045, A130233, A130256, A130260, A104162, A108852.
%Y A130234 Lucas inverse: A130241 - A130248.
%K A130234 nonn,easy
%O A130234 0,3
%A A130234 _Hieronymus Fischer_, May 17 2007