Q:

Distinct determinants of circulants

If $M$ is a circulant integer matrix of size $n\times n$ whose entries are randomly chosen from $\{0,1\}$ value, how many different determinants does $M$ possibly take value in? 
For $n=1,2,3,4$, I have atmost $1$ or $2$ different absolute values (signed value is atmost $3$).  For $n=5$, number seems $5$ different absolute values (signed value is atmost $9$).  I am suspecting $O(n)$ different value. 
If $M$ is symmetric and circulant, then what is the count? 

A:

The exact number of distinct absolute values of determinants  for $n = 1, \dots, 22$ is 
$2,2,3,3,5,6,9,11,15,19,23,59,55,111,223,258,361,880,1161,2285, 5075, 6041$
This is similar to loup blanc's list except for $a(13)$, $a(20)$ and $a(1)$.  I am not sure why we get different answers for $a(13)$
For matrices whose elements are chosen from $\{-1,1\}$ instead of $\{0,1\}$ the exact numbers for $n = 1, \dots, 24$ are:
$1, 1, 2, 2, 3, 2, 5, 5, 8, 5, 12, 24, 28, 25, 112, 87, 181,215, 581, 778, 2539, 1539, 8995, 20762$
Neither of these sequences are in OEIS.