A New Primes-Generating Sequence

For the sequence defined by $$a(n)=\frac{n^2-n-1}{\gcd (n^2-n-1,b(n-3)+nb(n-4))}$$ Where $b(n)=(n+2)(b(n-1)-b(n-2)),$ with initial conditions $b(-1)=0$ and $b(0)=1$, we find that $a(n)$ contains only $1$’s and primes, and can be represented as a finite continued fraction. It is more efficient for generating prime numbers than the Rowland sequence.