Newton's method for square roots 'jumps' through the continued fraction convergents

First some experimentation might be in order. I wrote up a program that could check for chains like this:

program nr
   implicit none
   integer, parameter :: ik16 = selected_int_kind(38)
   integer, parameter :: N = 200
   integer(ik16) p(N), q(N)
   integer D
   integer sqD
   integer r, s, a
   integer i, j, k
   integer(ik16) e, b, c

   write(*,'(a)',advance='no') 'Enter the value of D:> '
   read(*,*) D
   sqD = sqrt(D+0.5d0)
   r = 0
   s = 1
   a = (sqD+r)/s
   p(1) = a
   q(1) = 1
   r = a*s-r
   s = (D-r**2)/s
   a = (sqD+r)/s
   p(2) = a*p(1)+1
   q(2) = a
   do i = 3, N
      r = a*s-r
      s = (D-r**2)/s
      a = (sqD+r)/s
      p(i) = a*p(i-1)+p(i-2)
      q(i) = a*q(i-1)+q(i-2)
      if(p(i) <= p(i-1) .OR. q(i) <= q(i-1)) exit
   end do
   write(*,'(*(g0))') 'There are ',i-1,' convergents computed'
   outer: do j = 1, 5!(i-1)/2
      write(*,'(*(g0))') 'Starting convergent: ', j
      e = p(j)
      b = q(j)
      k = j
      do
         c = e**2+D*b**2
         if(c <= e) exit
         b = 2*e*b
         e = c
         do k = k+1, i-1
            if(b == q(k)) then
               if(e == p(k)) then
                  write(*,'(*(g0))') 'Matching convergent: ', k
                  exit
               else
                  write(*,'(a)') 'Matched denominator but not numerator'
                  stop
               end if
            else if(b < q(k)) then
               write(*,'(*(g0))') 'Missed convergent: ', k-1
               cycle outer
            end if
         end do
      end do
   end do outer
end program nr

So it always seemed to hit chains for $D=2$, $D=5$, $D=10$, $D=26$, $D=37$, and $D=50$ but for other values of $D$ sometimes it hit and sometimes it missed. My program has indices $1$ bigger than in the original question, so I'm seeing $n\rightarrow2n$ rather than $n\rightarrow2n+1$. Even the infamous $D=61$, which has a pretty large nontrivial solution to Pell's equation, has a chain. When another convergent is hit, it always seems to be exactly $2n$, never $2n-2$ or $2n+2$. It can't be $2n-1$ or $2n+1$ because Newton-Raphson approaches roots from above.

To prove a chain once found, it's probably useful to compare to solutoins to Pell's equation. Probably writing out the solutions in exponential form would lead to proof for a given chain.

EDIT: Just after posting it occurred to me that $$\begin{align}(p^2+Dq^2)^2-D(2pq)^2 & =p^4+2Dp^2q^2+D^2q^4-4Dp^2q^2\\ & =p^4-2Dp^2q^2+D^2q^4\\ & =(p^2-Dq^2)^2\end{align}$$ So every solution to Pell's equation $p^2-Dq^2=1$ leads to an infinite chain of solutions via Newton-Raphson and if $|p^2-Dq^2|\ne1$, then this metric grows exponentially so that the roots computed by Newton-Raphson will not be convergents.

EDIT: Relevant link to Newton-Raphson and Pell's equation.