Does every sequence $4^nx_0+\frac{4^n-1}{3}$ contain a prime?

Does every sequence $4^nx_0+\dfrac{4^n-1}{3}$ contain a prime?

Fix some odd positive integer $x_0$ and define the sequence $x_0,x_1,x_2,\ldots$ as follows:

$x_n=4^nx_0+\frac{4^n-1}{3}$

So for example choosing $x_0=1$ gives the sequence:

$1,5,21,85,\ldots$

And choosing $x_0=3$ gives the sequence:

$3,13,53,214,\ldots$

Does every such sequence contain a prime number?

Dirichlet's Theorem states that there are infinitely many primes in every arithmetic progression:

$a, a+d, a+2d,\ldots$

where $a,d$ are coprime. But I'm unaware of similar results for other progressions. I found this paper (which is beyond me) but my rudimentary understanding suggests this is saying the result is unknown for geometric progressions.

The progressions I'm asking about are exponential in nature. These are in fact the canonical set of linear combinations of the Lucas Sequences $U_1(5,4)$ and $V_1(5,4)$ over the odd integers.

Every successor $x_n:n\geq0$ is of course $\equiv 5\mod 8$ while every sequence for which $x_0\not\equiv\{1,3,7\}\mod 8$ is a subsequence of a longer sequence (with a smaller starting number), which does start with $x_0\equiv\{1,3,7\}\mod 8$.

So the set of all odd integers can be canonically indexed according to membership of the sequences starting with $x_0\equiv\{1,3,7\}\mod 8$.

I suspect these sequences have a close relationship with the kernel of the 2-adic logarithm, although I don't know a lot about that.

EDIT So far we have found that $5$ is the only prime in $1,5,21,85,...$ so not every sequence has infinitely many primes.

UPDATE Peter posted this related question for a candidate for a sequence with no primes and Paolo Leonetti has elegantly verified it has no primes, plus provided a class of such sequences which contain no primes; those for which $3x_0+1$ is a square number.

Aside of the nice solution provided in the other thread, which uses the algebraic factorization of the original formula when $3 x_0+1$ is a square, I have found a list of further solutions for $x_0$ such that the associated sequence contains no primes/only composites.
The sequences from $$x_0 \in \{371,495,775,1671,2171,3211,3955,4215,4655,4711,... \} \\ \text{have as well no prime entries.} \hspace{160pt} $$

This can easily be proved by looking at the small primefactors $p \in \{3,5,7,13\}$ : their cyclical occurences in the respective iteration sequence $S(x_0)$ have lengthes $\text{cyclen}(x_0,p) \in \{2,3,6\}$ and cover completely the $6$ leading entries of their sequence. Due to cyclicity each entry of the sequences $ \gt 13$ cannot be prime.

Example for $x_0=371$:

 N     primefact's x_N   Covering periodically
                          6 of 6 residueclasses
- - - - - - - - - - - - - - - - - - 
[ 0           "7.53"]       __7_        == 371
[ 1       "3^3.5.11"]       35__        == 4*371+1
[ 2         "13.457"]       ___1        == 4*(4*371+1)+1 
[ 3       "5.7^2.97"]       _57_  
[ 4        "3.31687"]       3___
[ 5      "5.113.673"]       _5__
- - - - - - - - - - - 
[ 6     "7.11.19753"]       __7_
[ 7    "3.5^2.81119"]       35__
[ 8    "13.293.6389"]       ___1
[ 9  "5.7.1427.1949"]       _57_
[10 "3^2.1423.30403"]       3___
[11  "5.11.28317907"]       _5__
- - - - - - - - - - - - 
 ... ... ... ...  ... 

Unfortunately I didn't find any further algebraical expression of this phenomenon, like the style of the one that has already been given with the $3x_0+1 \text{ -is-square}$ class of solutions.

Moreover there might be some further characteristics in/by the following list of $x_0$. The list has been created by checking the computing time up to $1000$ iterates, where the ones with $0$ msecs have been the suspect ones, and which gave the examples of the above mentioned covering-primefactors-solutions: $$ \quad \text{covering } \; \mid \; \text{ heuristics} \\ \small \begin{array} {rrr||rrr} x_0 & \# & \text{msecs}& x_0 & \# & \text{msecs}\\ \hline 371 & 0 & 0 & 35 & 0 & 125 \\ 495 & 0 & 0 & 603 & 0 & 171 \\ 775 & 0 & 0 & 1195 & 0 & 282 \\ 1671 & 0 & 0 & 1335 & 0 & 218 \\ 2171 & 0 & 0 & 1379 & 0 & 266 \\ 3211 & 0 & 0 & 1611 & 0 & 31 \\ 3955 & 0 & 0 & 1751 & 0 & 141 \\ 4215 & 0 & 0 & 2755 & 0 & 172 \\ 4655 & 0 & 0 & 2775 & 0 & 234 \\ 4711 & 0 & 0 & 2791 & 0 & 156 \\ 4971 & 0 & 15 & 3179 & 0 & 250 \\ 5195 & 0 & 0 & 3379 & 0 & 140 \\ 5831 & 0 & 16 & 4011 & 0 & 204 \\ 5955 & 0 & 0 & 4235 & 0 & 94 \\ 6235 & 0 & 0 & 5395 & 0 & 141 \\ 7131 & 0 & 0 & 5495 & 0 & 47 \\ 7631 & 0 & 0 & 5751 & 0 & 93 \\ 8671 & 0 & 0 & 5975 & 0 & 125 \\ 9415 & 0 & 0 & 6215 & 0 & 93 \\ 9675 & 0 & 0 & 6571 & 0 & 141 \\ 10115 & 0 & 0 & 6671 & 0 & 140 \\ 10171 & 0 & 0 & 6875 & 0 & 126 \\ 10431 & 0 & 0 & 7551 & 0 & 171 \\ 10655 & 0 & 0 & 7775 & 0 & 203 \\ ... & ... & ... & ... & ... & ... \end{array}$$ where "#" denotes the number of primes found up to $1000$ (second run $8000$) iterations.
The timings were measured for up to the initial $N=1000$ iterations (using primecertifying isprime() in Pari/GP vers. 2.14, 64 bit.

updated table:The column headed with "covering" contains that cases of $x_0$ for which the covering by the small primes occurs, so this gives proven non-prime sequences, the column headed with "heuristics" contains the cases, where the primecertification showed all iterates being composite but only up to $N=8000$ iterations.

For instance, $x_0=35$ needs further work to discern whether the iterations sequence has primes or not; there have been a couple of prime-canditates by a fast (pseudo)prime-test for $N>1000$ but which were certificated to be composite with the more intense isprime()-function. The time to do this increases excessively and unpredictable, so I stopped testing of $x_N$ at $N=8000$.