A strange "pattern" in the continued fraction convergents of pi?

A strange "pattern" in the continued fraction convergents of pi?

From the simple continued fraction of $\pi$, one gets the convergents,
$$p_n = \frac{3}{1}, \frac{22}{7}, \frac{333}{106}, \frac{355}{113},
\frac{103993}{33102}, \frac{104348}{33215}, \frac{208341}{66317},
\frac{312689}{99532}, \frac{833719}{265381}, \frac{1146408}{364913},
\dots$$
starting with $n=1$, where the numerators and denominators are A002485 and
A002486, respectively. If you stare at it hard enough, a pattern will
emerge between three consecutive convergents. Define,
$$a_n,\,b_n,\,c_n = p_{n}-3,\;\; p_{n+1}-3,\;\; p_{n+2}-3$$
$$v_n=\text{Numerator}\,(a_n)\,\text{Numerator}\,(b_n)$$
then for even $n \ge 2$,
$$F(n) = \sqrt{\frac{a_n c_n}{a_n-c_n}-v_n}=\text{Integer}\, (often)$$
For example, for $n = 2$,
$$a_2,\,b_2,\,c_2, = \frac{22}{7}-3,\; \frac{333}{106}-3,\;
\frac{355}{113}-3$$
$$F(2) = 1$$
More generally,
$$\begin{array}{cc} n&F(n) \\ 2&1 \\ 4&16\\ 6&4703\\ 8&14093\\ 10&51669\\
12&122126\sqrt{2}\\ 14&7468474\\ 16&\frac{18549059}{\sqrt{2}}\\
\end{array}$$
and so on. For even $n<100$, I found half of the $F(n)$ were either
integer or half-integer. (And all the non-integers were of form
$N\sqrt{d}$ for some very small d.)
Some questions:
For $n<500$, $n<1000$, etc, how many $F(n)$ are integers or half-integers?
More importantly, why is $F(n)$ often an integer?