On the continued fraction expansion of the unique root in F(p) of the equation x4+x2−Tx−1/12=0 and other related hyperquadratic expansions

In 1986, Mills and Robbins observed by computer the continued fraction expansion of certain algebraic power series over a finite field. Incidentally, they came across a particular equation of degree four in characteristic p=13. This equation has an analogue for all primes p⩾5. There are two patterns for the continued fraction of the solution of this equation, according to the residue of p modulo 3. We describe this pattern in the first case. In the second case we only give indications.