Calculating nth element of two sequences dependent on each other
Solve for $b_{n-2}$ in the first equation to get
$$b_{n-2}=a_n-a_{n-1}+1$$
Reindex and substitute $b_n$ and $b_{n-2}$ into the second equation to get
$$a_{n+2}-a_{n+1}+1=2a_n+a_n-a_{n-1}+1+4$$
Solve for $a_{n+2}$ to get
$$a_{n+2}=a_{n+1}+3a_n-a_{n-1}+4$$
Reindex to get
$$a_{n}=a_{n-1}+3a_{n-2}-a_{n-3}+4$$
To use this, we need $a_0$, $a_1$, and $a_2=a_1+b_0-1$.