Let $F=\mathbb{Q}\left(\sqrt{d}\right)$ be a real quadratic field with ring of integers $𝒪$. In this paper we analyze the number $h_d$ of ${GL}_1\left(𝒪\right)$-orbits of homothety classes of perfect unary forms over $F$ as a function of $d$. We compute $h_d$ exactly for square-free $d\le 200000$. By relating perfect forms to continued fractions, we give bounds on $h_d$ and address some questions raised by Watanabe, Yano, and Hayashi.

In this paper we consider proper cycles of indefinite integral quadratic forms $F=(a,b,c)$ with discriminant $\Delta$. We prove that the proper cycles of $F$ can be obtained using their consecutive right neighbors $R^i\left(F\right)$ for $i\ge 0$. We also derive explicit relations in the cycle and proper cycle of $F$ when the length $l$ of the cycle of $F$ is odd, using the transformations $\tau \left(F\right)=(-a,b,-c)$ and $\chi \left(F\right)=(-c,b,-a)$.