Let $A$ be the algebra corresponding to a representation-finite block of a Schur algebra. See for example 6.1. of https://arxiv.org/pdf/1607.05965.pdf for quiver and relations and some relevance of this algebra. Define the permanent of an algebra as the permanent of the Cartan matrix of that algebra (Not sure if someone considered that before, I just played a little with it). The algebra $A$ is for example famous for having a double centraliser property with the symmetric group (Schur-Weyl duality). In this case of representation-finite $A$ one has a minimal faithful projective-injective module $eA$ ($e$ an idempotent of $A$) such that $eAe$ is isomorphic to a block of the group algebra of the corresponding symmetric group. Let $B$ be this block. Now for the sequence $a_n$ of permanents of this block $A$ of the Schur algebra with $n$ simple modues i obtained the sequence of numerators of continued fractions convergent to $\sqrt{2}$, see https://oeis.org/A001333. For the sequence $b_n$ of permanents of the corresponding algebra $B$ via Schur-Weyl duality I got the sequence of denominators of continued fraction convergent to $\sqrt{2}$, see https://oeis.org/A000129.
(Here I mean the denominator and numerators of the sequence $a_{n+1}=\frac{a_n}{2}+\frac{1}{a_n}$ with $a_0=1$, which has limit $\sqrt{2}$.)
This raises more general questions:
-Is there a good explanation for this expect calculations? Does something like this also appear for other Schur algebras and the corresponding symmetric group? (or other algebras)
-What are the permanents of the (blocks of) group algebra of the symmetric group or the Schur algebras (when they are not semisimple) in general? Do they have some kind of meaning?
-Is there more behind this? Maybe something like this works more general for algebras and modules (possible minimal faithful projective-injective) having the certain double centraliser properties?
