Newton's approximants and continued fraction expansion of 1+√d/2

Let d be a positive integer such that d equivalent to 1 (mod 4) and d is not a perfect square. It is well known that the continued fraction expansion of 1+root d/2 is periodic and symmetric, and if it has the period length l <= 2, then all Newton's approximants R-n = p(n)(2) + d-1/4 q(n)(2)/q(n)(2p(n) - q(n)) are convergents of 1+root d/2 and then it holds R-n = p(2n+1)/q(2n+1) for all n >= 0. We say that R-n is a good approximant if R-n is a convergent of 1+root d/2. When l > 2, then there is a good approximant in the half and at the end of the period. In this paper we prove that being a good approximant is a palindromic and a periodic property. We show that when l > 2, there are R-n's, which are not good approximants. Further, we define the numbers j = j(d, n) by R-n = p(2n+1+2j)/q(2n+1+2j) if R-n is a good approximant; and b = b(d) = \{n : 0 <= n <= l - 1 and R-n is a good approximant}\. We construct sequences which show that \j\ and b are unbounded.