QT-blocks
QT-blocks involve the other two fabfour series that are always part of the set of series of a fractional approach, the quartus and the tertius.
We've defined a fraction's value 'v' = (numerator)2-n*(denominator)2. With regard to this, qt-blocks come in three kinds:
  1. A multiple of the tertius in the numerator, a multiple of the quartus in the denominator.
    In this case v = t2-nq2 > 1 and the value of the fraction is above the root.
  2. A multiple of the quartus in the numerator, a multiple of the tertius in the denominator.
    In this case v = q2-nt2 < 0 and the value of the fraction is below the root.
  3. Both of the above in successive fractions.
    In this case the t/q fraction precedes the q/t fraction and v1*v2 = -n.
    4.
The value of 'v' is the same for all qt-blocks in a particular series.
For carréphobic numbers, all fractions involved are indeed multiples of the tertius and quartus.
Carréphylic numbers on the other hand always have a single tertius or quartus in the denominator.

From the first qt-block ⇒ the first non trivial sp-block
The following formulas hold for both carréphylic and carréphobic numbers. For the former they are inconsequential because the diophantine equation s2-np2 = 1 can be solved. For the latter they allow the calculation of 's' and 'p' in cases where a direct search fails.
As far as this investigation is concerned, I used them to determine the sp-blocks of √61 (1766319049 / 226153980) and √109 (158070671986249 / 15140424455100), both of which were out of range of Ed's fractional approximations program.

Conjecture
  • A fraction is part of a (semi-)symmetric series if and only if v | 2*(numerator).
    If v = 1 we have the first fraction of a sp-block, otherwise it is a qt-block.
Let's take a t/q fraction, so v > 1 and v | 2t. Since t2-nq2= v, the distance of nq2 to the nearest square (t2) is v, which divides 2t. Hence nq2 is carréphylic, so for the sp-series linked to the approach of its root: s = (2t2-v)/v and p = 2t/v.
Since √nq2 = Q√n, s/p is an approximation of Q√n, so for √n itself:
  • s = (2t2-v)/v and p = 2qt/v
A similar reasoning applies to q/t fractions:
  • s = -(2q2-v)/v and p = -2tq/v
 
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