I think this is rather a naive question, so please forgive me for asking this.
In the Wikipedia page List of statements independent of ZFC, it is stated that "One can write down a concrete polynomial $p\in\mathbb{Z}[x_1,\cdots,x_9]$ such that it can neither be proven nor disproven in $\mathsf{ZFC}$ that $p$ has a zero in $\mathbb{Z}^9$ (assuming $\mathsf{ZFC}$ is consistent)." Since this is a polynomial written out concretely, the polynomial does not depend on the concrete model of $\mathsf{ZFC}$; what's astonishing is that the question whether the polynomial has a zero does, however.
So I was curious about if there is a similar phenomenon for the integer sequences. Is there a sequence in $\mathbb{N}^*$, not depending on the concrete model of $\mathsf{ZFC}$, such that $\mathsf{ZFC}$ neither proves nor disproves its convergence? A sequence like $a_{2n}=0$, $a_{2n-1}=\begin{cases} 1, \text{if }\mathsf{CH}\text{ holds}\\ 0, \text{if not}\end{cases}$ depends on the model, so it is not what I am looking for. Thank you for any help in advance, and please don't hesitate to correct me for any mistakes. In addition, if such a sequence exists, a few terms provided whenever possible would be appreciated.
Enumerate all the tuples in $\mathbb{Z}^9$ and define $a_n:=n$ if there is $i<n$ such that the $i$-th tuple is a zero for the fixed polynomial $p$ you described, and $a_n:=0$ otherwise. Then this sequence converges iff $p$ does not have a zero.