Little Explanation
here is a recurrence relation defined by $d_n = 3d_{n - 2} - d_{n - 3}$ where $d_1=0, d_2 = 1$ and $d_3 = 2$. this recurrence relation can be used to test if number $p$ is prime. and $p$ is prime if and only if $d_p, d_{p+1}$ and $d_{p+2}$ are either $\{1, 2, 3\}$, $\{2, 1, 1\}$ or $\{-3, -3, -4\}$ all of these $\bmod{p}$. this test is deterministic for all number, $n$ such that $n\not\equiv\pm{1}\bmod{18}$, since all pseudoprimes are of this form, why? this seuence is periodic and the period of all prime numbers that are congruent to $\pm{1}\bmod{18}$ is $(p - 1)\times{k}$ where $k$ is less than or equal to $1$. this kind of period that depend on $p - 1$ cause periodic collision resulting to pseudoprimes. and all pseudoprimes have all of its prime factors of form $\pm{1}\bmod{18}$ but i havent found any pseudoprime of congruent to $-1\bmod 18$ but i suspect they exist.
All prime numbers $p$ that are not congruent to $\pm1\bmod 18$ have periods of form ${2(p^2 + p + 1)\dot{k^{-1}}}$, these periods explodes to very big number and this avoids periodic collision that may result to pseudoprimes.
If you want to think about pseudoprimes with respect to third-order recurrence sequences, a good place to start is Adams and Shanks' 1982 paper on Perrin pseudoprimes. I suspect you will find that all of your examples are what they call "S-pseudoprimes". There are probably infinitely many "Q-pseudoprimes", but nobody has found one yet.