I am interested in the 1st solutions to this set of equations, and wonder if there are any techniques I could use to try and yield an nth term. I'll provide the first few for clarity.
General Equation:
$$3^aN_a + 3^a - 2^a ≡ 2^{2×3^{a-1}} \; mod \; 2^{2×3^{a-1}+1}$$
When a=1
$$ 3N_1 + 1 ≡ 4 \;mod\;8 \;\;\;\;⇒\;\;\;N_1=1$$
When a=2
$$ 9N_2 + 5 ≡ 64 \;mod\; 128 \;\;\;\;⇒\;\;\;N_2=35$$
When a=3
$$ 27N_3 + 19 ≡ 262144 \;mod\; 524288 \;\;\;\;⇒\;\;\;N_3=184471$$
To clarify, I am interested in the sequence: $N_1, N_2, N_3, ...$ and I'm aiming for something of the form: $N_a = \;\;?$
Thank you in advance for any assistance.