Sequence defined by polynomial recursion

Question

I'm struggling with a sequence that is defined by recursion, and that looks like : n_t+1 = n_t + C/n_t where C is a constant. How should I proceed to find the general formula for any term n_t, depending on n₀ and t ?

Answer 1

Opening the sequence partially shows that $$n_{t+1}=n_t+\frac{C}{n_t}=n_{t-1}+\frac{C}{n_{t-1}}+\frac{C}{n_t}=n_{t-2}+\frac{C}{n_{t-2}}+\frac{C}{n_{t-1}}+\frac{C}{n_{t}}+\dots$$
$$\dots=n_0+C\sum_{k=0}^t\frac{1}{n_k}$$
where $n_0$ is some initial value $c_0$. Also assuming that $\sum_{k=0}^t\frac{1}{n_k}$ is limited.

EDIT: General formula is still unknown, since closed form for the sum does not exists. On the other hand, value of arbitrary (limited) $n_t$ is easily computed with program of several codelines if $C$ and $n_0$ are defined.