A130260 - OEIS

%I #19 Sep 08 2022 08:45:30

%S 0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,

%T 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,

%U 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6

%N Minimal index k of an even Fibonacci number A001906 such that A001906(k) = Fib(2k) >= n (the 'upper' even Fibonacci Inverse).

%C Inverse of the even Fibonacci sequence (A001906), since a(A001906(n))=n (see A130259 for another version).

%C a(n+1) is the number of even Fibonacci numbers (A001906) <=n.

%H G. C. Greubel, <a href="/A130260/b130260.txt">Table of n, a(n) for n = 0..10000</a>

%H Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, <a href="https://doi.org/10.2478/auom-2021-0002">On some new results for the generalised Lucas sequences</a>, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36.

%F a(n) = ceiling(arcsinh(sqrt(5)*n/2)/(2*log(phi))) for n>=0.

%F a(n) = ceiling(arccosh(sqrt(5)*n/2)/(2*log(phi))) for n>=1.

%F a(n) = ceiling(log_phi(sqrt(5)*n)/2)=ceiling(log_phi(sqrt(5)*n-1)/2) for n>=1, where phi=(1+sqrt(5))/2.

%F a(n) = A130259(n-1) + 1, for n>=1.

%F G.f.: g(x)=x/(1-x)*Sum_{k>=0} x^Fib(2*k).

%e a(10)=4 because A001906(4)=21>=10, but A001906(3)=8<10.

%t Join[{0}, Table[Ceiling[Log[GoldenRatio, Sqrt[5]*n]/2], {n, 1, 100}]] (* _G. C. Greubel_, Sep 12 2018 *)

%o (PARI) for(n=0,100, print1(if(n==0, 0, ceil(log(sqrt(5)*n)/(2*log((1+ sqrt(5))/2)))), ", ")) \\ _G. C. Greubel_, Sep 12 2018

%o (Magma) [0] cat [Ceiling(Log(Sqrt(5)*n)/(2*Log((1+ Sqrt(5))/2))): n in [1..100]]; // _G. C. Greubel_, Sep 12 2018

%Y Cf. partial sums A130262. Other related sequences: A000045, A001519, A130234, A130237, A130239, A130256, A130259. Lucas inverse: A130241 - A130248.

%K nonn

%O 0,3

%A _Hieronymus Fischer_, May 25 2007, May 28 2007, Jul 02 2007