Can we always find values $(a,d)$ such that $\frac{d+an}{4a-1}$ is an integer for all $n$ I'm working on determining if a certain form of simple hyperbolic diophantine equation always has solutions. I did some work and determined the answer is yes if the following fraction can always be an integer for some $a$ and some $d$ dividing $(an)^2$.

For all $n$, can we always find values $(a,d)$ such that $\dfrac{d+an}{4a-1}$ is an integer and where, 
  
  
*
  
*$n$ is any fixed integer $\geq2$
  
*$a$ is any positive integer
  
*$d$ is a positive divisor of $(an)^2$.
  
  
  Required: One need to find at least one couple $(a,d)$ per value of $n$ verifying the conditions.

I have checked for values of $n$ up to one million using a computer. Does anybody have any ideas how to show this is true or have a counterexample? 
By the way, this purely recreational. Thanks for any help or ideas!
Edit: Sorry for the confusion. I only care that some $a$ and some $d$ dividing $(an)^2$ exist such that the fraction becomes an integer. But $a$ can take on any positive integer value.
Update: I have made some progress with the problem. I can show the statement holds for all integers not of the form $n=24k+1$. Furthermore, I can show that if the statement holds for a composite integer's prime factors, it holds for the composite number itself. So, I have narrowed the problem down to primes $n=p=24k+1$. Hopefully somebody has an idea to deal with the primes.
 A: I have a partial result.
For $n=3q+r$ with $r\in\{0,1,2\}$
We have for $a=1$
$$F=\dfrac{an+d}{4a-1}=\dfrac{3q+r+d}3=q+\dfrac{d+r}3$$



*

*For $r=0$ then $d=3\qquad$ [ $n^2=(3q)^2$ so $d\mid n^2$ ]

*For $r=2$ then $d=1\qquad$ [ $d=1$ always divides $n^2$ ]

*For $r=1$ then $d=2$ and $q\text{ odd}\qquad$ [ $n^2\equiv q+1\pmod{2}$, so if $q$ odd $d=2$ divides $n^2$ ]
In all three cases then $d+r=3$ so $F$ is an integer.

The only difficult case appears to be when $n=6p+1$. 
For that case, the simulation on computer does not show a good pattern for $a$ or $d$.

Edit 28/11
Inspired by mathlove post I took my chance at endings $1,9$ in decimal or 
or $k=5m,5m+2$.
The most promising road was $\begin{cases}a=uv\\d=un\end{cases}$ 
Resulting in $\dfrac{u(1+v)}{4uv−1}\in\mathbb N$. 
Unfortunately the numerator is too small and this doesn't work out. 
Then I explorer another idea with computer simulation $\begin{cases}a=uv\\d=uan^2\end{cases}$
In this case $f(a,d,n)=\dfrac{an+d}{4a-1}=an\times\dfrac{1+un}{4uv-1}=an\times g(u,v,n)$.
Still for the $n=24k+1$ primes, this reduced problem $g(u,v,n)\in\mathbb N$ seems almost always to have a solution in $(u,v)$
Only very few cases actually do not work (either impossible or u,v were possibly too large).
$k\in\{8,105,2772,3185,10500,13217,13728,16875,50547,59892,72140,81593,83252,...\}$
I don't know what particularity these numbers share, maybe it is a good place to dig in, or maybe it also means that we have to go for another approach. Anyway, if anyone is tempted to explore further, this is what to expect.
