PRNG that takes input a $n$-bit string and gives output a $2n$-bit string?
It can be consctructed in many different ways.

I will just give an example using HMAC-$\mathcal{H}$ where $\mathcal{H}$ is a hash function which returns a $x$-bit hash.

So if you want to use a PRNG which takes $n$-bit inputs and returns $2n$-bit output, you could act this way:


Divide the input into $\frac{x}{2}$-bit blocks $B_i$
For each block $B_i$ computes $B'_i = \text{HMAC-}\mathcal{H}_K(B_i \space || \space i)$ where $K$ is the symmetric key used for the HMAC computation and $||$ denotes concatenation.


In that case, note that the property $n$-bit input returns a $2n$-bit output is confirmed only if $n \mod \frac{x}{2} = 0$, but you can still truncate the result after HMAC computations to get the expected output length.