edit: I added conjecture 2 that looks much more accessible.


Here is the elementary combinatorial translation of the problem (read below for the homological background):


Let $n \geq 2$.
A Nakayama algebra is a list $[c\_0,c\_1,...,c\_{n-1}]$ of $n$ integers with $c\_i \geq 2$, $c\_{i+1} \geq c\_i-1$ and $c\_{n-1}=c\_0+1$ (it can be interpreted as a periodic Dyck path). Read the indices $i$ in the $c\_i$ mod $n$ so that they are defined for any $i \in \mathbb{Z}$.
A module is a tuple $(i,k)$ with $0 \leq i \leq n-1$ (the first entry $i$ is always viewed mod $n$) and $1 \leq k \leq c\_i$. A proper module is a module with $k < c\_i$.
A proper module $M=(i,k)$ is called weird in case with $k=s+tn$ with $0 \leq s <n$ we have $n \leq k$ and $max(0,2k-c\_i) \leq s+(t-1)n$ (this should be equivalent to $Ext\_A^1(M,M) \neq 0$ ).
Define the $i$-th syzygy of a proper module $(i,k)$ as follows:
$\Omega^0(i,k)=(i,k)$, $\Omega^1(i,k)=(i+k,c\_i-k)$ and $\Omega^l(i,k)=\Omega^1(\Omega^{l-1}(i,k))$ for $l \geq 2$.
A proper module $M=(i,k)$ is called periodic with period $q$ in case $M = \Omega^q(M)$ and it is called quasi-periodic in case $\Omega^u(M)$ is periodic for some $u \geq 0$.


The below guess can now be stated completely elementary (I state it here as a conjecture, but it is hard to check since there are infinitely many Nakayama algebras for a given $n$):



> 
> Conjecture: Any weird module is quasi-periodic.
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> 
> Conjecture 2: In case $M$ is weird, also $\Omega^1(M)$ is weird.
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Conjecture 2 implies conjecture 1 and looks much easier but I cant find a good argument yet.


Example: The Nakayama algebra [3,4] has the unique weird module (1,2) , which is periodic of period 1.


Homological background:


The strong no loop conjecture states that for a simple module $S$ over a finite dimensional algebra $A$ we have that $S$ has infinite projective dimension in case $Ext\_A^1(S,S) \neq 0$. In general this is not true for arbitrary indecomposable modules instead of simple modules, but it seems that it might be true for Nakayama algebras.



> 
> Guess: In case $A$ is a Nakayama algebra with an indecomposable module $M$ with $Ext\_A^1(M,M) \neq 0$ then $M$ has infinite projective dimension.
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> 
> 


I can prove this for algebras with finite global dimension (which includes all Nakayama algebras with a linear quiver and thus we can focus on cyclic quivers).
I should note that maybe there is an easy argument and Im too blind to see it at the moment.



Combinatorial problem on periodic dyck paths from homological algebra
I think you have a typo in the definition of a Nakayama algebra list and it should read "$c\_{i+1}\geq c\_i-1$." If this is the case then conjecture 2 has a simple proof:


The condition for a module $M=(i,k)$ to be weird is that $n\le k\le c\_i-n$. We have to show that this implies that $\Omega^1(M)=(i+k,c\_i-k)$ is also weird. This means that we have to check that
$$n\le c\_i-k\le c\_{i+k}-n.$$
Now, the inequality $n\le c\_i-k$ follows immediately from the weirdness of $M$. The second inequality follows from observing that
$$c\_{i+k}-c\_i=c\_{i+k}-c\_{i+n}=\sum\_{j=n+1}^{k}(c\_j-c\_{j-1})\geq \sum\_{j=n+1}^k (-1)=n-k.$$
After rearranging the terms we get $c\_i-k\le c\_{i+k}-n$ as desired.


