We have a $n \times n$ matrix $a_{ij} = a^{|i-j|}$ which looks like this:
$$A=\pmatrix{1&a&a^2&\cdots&a^{n-1}\\ a&1&a&\cdots&a^{n-2}\\ a^2&a&1&\cdots&a^{n-3}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ a^{n-1}&a^{n-2}&a^{n-3}&\cdots&1}$$
Computing the first few determinants yields:
$\det(A_1)=1$
$\det(A_2)=1-a^2$
$\det(A_3)=a^4-2a^2+1$
$\det(A_4)=-a^6+3a^4-3a^2+1$
Is there a pattern?
