A130239 - OEIS

%I #17 Mar 18 2023 03:56:33

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

%T 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,

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

%N Maximal index k of the square of a Fibonacci number such that Fib(k)^2 <= n (the 'lower' squared Fibonacci Inverse).

%H G. C. Greubel, <a href="/A130239/b130239.txt">Table of n, a(n) for n = 0..5000</a>

%F a(n) = max(k | Fib(k)^2 <= n) = A130233(floor(sqrt(n))).

%F a(n) = floor(arcsinh(sqrt(5n)/2)/log(phi)), where phi=(1+sqrt(5))/2.

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

%e a(10) = 4 since Fib(4)^2 = 9 <= 10 but Fib(5)^2 = 25 > 10.

%t A130233[n_]:= Floor[Log[GoldenRatio, 3/2 +n*Sqrt[5]]];

%t Table[A130233[Floor[Sqrt[n]]], {n, 0, 120}] (* _G. C. Greubel_, Mar 18 2023 *)

%o (Magma)

%o A130233:= func< n | Floor(Log(3/2 + n*Sqrt(5))/Log((1+Sqrt(5))/2)) >;

%o [A130233(Floor(Sqrt(n))): n in [0..120]]; // _G. C. Greubel_, Mar 18 2023

%o (SageMath)

%o def A130233(n): return int(log(3/2 +n*sqrt(5), golden_ratio))

%o def A130239(n): return A130233(floor(sqrt(n)))

%o [A130239(n) for n in range(121)] # _G. C. Greubel_, Mar 18 2023

%Y Partial sums: A130240. Other related sequences: A000045, A130233, A130234, A130235, A130236, A130237, A130238, A130240, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

%K nonn

%O 0,2

%A _Hieronymus Fischer_, May 17 2007, May 28 2007