According to Theorem 5.1 of
http://math.mit.edu/~rstan/pubs/pubfiles/35.pdf, the number is
  $$ \frac 1p\left[ 2^{p-1}+\frac{p-1}{2}(\epsilon^{4h} +
      \epsilon^{-4h})\right], $$
where $h$ is the class number of $\mathbb{Q}(\sqrt{p})$ and
$\epsilon>1$ the fundamental unit of $\mathbb{Q}(\sqrt{p})$.