For $n \geq 2$ let $B_n$ be the algebra of upper triangular matrices over a field $K$. Recall that two algebras are said to be stably equivalent in case their module categories modulo projectives are equivalent as categories. Let $b_n$ denote the number of connected finite dimensional algebras stably equivalent to $B_n$.
Question:
Is there a nice description of algebras stably equivalent to $B_n$? And what is the sequence $b_n$?
Note that the stable category of $B_n$ seems to be quite special since it is an abelian category (I do not know many other examples where this is the case).
Im not so familiar with stable equivalences so Im sorry in case this question is not appropriate for MO, but it seems to me that for larger $n$ this might not be so easy although it feels like the sequences are very small (but I have no good argument for that).
(edit1: The question about $A_n=K[x]/(x^n)$ was removed. For $n \geq 3$, $A_n$ should only be stably equvialent to itself, since all indecomposable non-projectives have non-zero stable Homs between them and thus an algebra stably equivalent to $A_n$ must be local since stable Homs between non-isomorphi simple modules are zero and the only other possibility has too many indecomposable non-projectives.)
