<s>[INST] Continuous polar equation of Sqrt spiral Please help find polar coordinate equation  $ r =f(\theta) $ or $ \theta =g(r) $ 
where $r$ and $\theta$ are continuous functions of  $n$ as given: 
SqrtSpiral
EDIT1:
n is a discrete (discontinuous) variable ( n = 1,2,3..) with
$$ r_n = \sqrt n ;$$
$$ r_{n+1}^2 -r_n^2 = 1 ;  \\ r_n = \cot\Delta\theta =\cot (\theta_{n+1} - \theta_n );$$
All known spirals in the plane are either 
$ r= f(\theta) $ or parametrically $ r(t),\theta(t), $ 
into which form Sqrt spiral is not castable.
I tried to adjust Pythogorean relation
$$ m^2+n^2 , m^2-n^2 , 2 m n $$
with no luck, so I posted here for help.
 [/INST] $r_{n+1}^2 -r_n^2 = 1 ;  \\ r_n = \cot\Delta\theta =\cot (\theta_{n+1} - \theta_n ) ;$
I'll write $t$ for $\theta$ since I'm lazy.
From your equations,
$\cot^2 (t_{n+2} - t_{n+1} ) 
=\cot^2 (t_{n+1} - t_n ) +1
$,
so
$t_{n+2}$
is a function of
$t_{n+1}$
and
$t_{n}
$.
Since
$\cot(a-b)
=\dfrac{1+\tan a \tan b}{\tan a -tan b}
$,
letting
$u_n
=\tan(t_n)
$,
$\dfrac{(1+u_{n+2} u_{n+1})^2}{(u_{n+2}-u_{n+1})^2}
=\dfrac{(1+u_{n+1} u_{n})^2}{(u_{n+1}-u_{n})^2}+1
$.
This is the relation among
the $u_i$.
Letting 
$g
=\dfrac{(1+u_{n+1} u_{n})^2}{(u_{n+1}-u_{n})^2}+1
$,
$(1+u_{n+2} u_{n+1})^2
=g(u_{n+2}-u_{n+1})^2
$
or,
letting
$p = u_{n+2}$
and
$q = u_{n+1}$,
$(1+pq)^2 = g(p-q)^2$
or
$1+2pq+p^2q^2
=g(p^2-2pq+q^2)
$
or
$p^2(q^2-g)
+p(2-2gq)
+gq^2-1
=0
$,
or
$p^2(q^2-g)
+2p(1-gq)
+gq^2-1
=0
$.
The discriminant of this
divided by $4$
 is
$\begin{array}\\
((1-gq)^2-(q^2-g)(gq^2-1)
&=(1-2gq+g^2q^2)-(gq^4-q^2(g^2+1)+g)\\
&=(1-2gq+g^2q^2)-(gq^4-q^2(g^2+1)+g)\\
&=1-2gq+g^2q^2-gq^4+q^2g^2+q^2-g\\
&=1-2gq+2g^2q^2-gq^4+q^2-g\\
\end{array}
$
I don't know where to go from here,
so I'll stop.
</s>