BG-JBMO-TST 2022 Problem 3

For a positive integer $n$ let $t_n$ be the number of unordered triples of non-empty and pairwise disjoint subsets of a given set with $n$ elements. For example, $t_3 = 1$ . Find a closed form formula for $t_n$ and determine the last digit of $t_{2022}$ .

(I also give here that $t_4 = 10$ , for a reader to check his/her understanding of the problem statement.)