Given $N_1=26,~ N_2=20,~ N_3=4,~ N_4=16,~ N_5=10,~ N_6=2,$ and $N_7=14,$ and working backwards from
>!$N_8=8,$

which [f'' found somewhere](/questions/36655/what-is-the-next-number-is-this-sequence#comment108625_36655), I can reverse-engineer

>!\begin{align}N_1&=26~~\text{(seed value)}&N_4&=N_3+12&N_7&=N_6+12\\N_2&=N_1-6&N_5&=N_4-6&N_8&=N_7-6\\N_3&=N_2-2^4&N_6&=N_5-2^3&…\end{align}

… but that isn’t very satisfying.  You could probably construct
equally plausible derivations for lots of other answers.