“middle” partial denominator in continued fraction expansion of square roots

Suppose d is a positive integer that is not a perfect square such that the negative Pell equation, x2dy2=1 has no solution. Then we know the minimal period of the continued fraction expansion of d has even length, 2, and that the partial denominators, a1,,a21, are symmetric about the -th partial denominator, a. I.e., aj=a+j for j=1,,1. It is this partial denominator, a, that I am referring to here as the middle partial denominator.

It is a classical result that a2=2a0, but my question is what is known about the middle partial denominator, a, under the above conditions on d.

Of course, if d=a20+2, where a0 is a positive integer, for example, then we know that the continued fraction expansion of d takes the form [a0;¯a0,2a0], so the middle partial denominator is a0 here. But are there more general results known that do not depend on d satisfying such quadratic expressions that have “nice” continued fraction expansions?

Any known results with references, ideas,… would be greatly appreciated.

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