  Suppose that $p$ is an odd prime and $\genfrac{(}{)}{}{}{\cdot}{p}$ denotes
the Legendre symbol modulo $p$. If $p$ is has the form $p= n^2+1$ then one
easily verifies that $\genfrac{(}{)}{}{}{a}{p} = \genfrac{(}{)}{}{}{-a}{p}$ for
all $a\in \mathbb Z/p\mathbb Z$. We identify various symmetry properties of
sequential matrices over $\mathbb Z/(n^2+1)\mathbb Z$ regardless of whether
$n^2+1$ is prime. We deduce from these results a collection of symmetries
involving Jacobi symbol modulo $n^2+1$ which generalize our above observation
on the Legendre symbol.
