The following is a definition of a Banach space that is a generalization of the original Tsirelson space. Nowadays such a space is called a Mixed Tsirelson space; it was introduced by Argyros and Deliyanni in 1997.
It is clear that, for fixed $x \in c_{00}$, the sequence $( \lvert x \rvert_n )_{n \in \mathbb{N}}$ is increasing and bounded above by the $\ell_1$-norm of $x$; so that $\lVert x \rVert$ is well-defined.
I think that this sequence stabilizes from $k := \lvert \text{supp}(x) \rvert$ onwards, i.e.,
$$\lvert x \rvert_k = \lvert x \rvert_{k+1} = \lvert x \rvert_{k+2} = \cdots.$$
Is that the case?
It's been a couple of days now, and I haven't received any answer so far. The following is, I believe, a proof of this fact. I'm posting the required definitions from my notes, as well. Any comments would be appreciated.