Is there an easy way to solve a recurrence given with two variables and three different functions? Actually I'm looking for the solution of:
$$A(n,k)=A(n-2,k-1)+A(n-3,k-1)+R(n-2,k-1)+L(n-2,k-1) $$ $$R(n,k)=A(n-1,k-1)+ R(n-1,k-1)$$ $$L(n,k)=A(n-1,k-1)+ L(n-1,k-1)$$
For the initial value I have: $A(1,0)=1$, $L(0,0)=1$, $R(0,0)=0$ and: $$ A(n,k)=0 \text{ if }k+1>n \text{ or } n>3*k+1$$ $$ R(n,k)=0 \text{ if }k+1>n \text{ or } n>3*(k-1)+2$$ $$ L(n,k)=0 \text{ if }k>n \text{ or } n>3*(k-1)+2$$
I have no clues of where to start ... Any idea or even (free) tools would be welcome.
Thanks.
I edited because may inequality where mixed up. Sorry. It should be correct now I also addes some initial value for R and L. Not sure it's necessary but it cannot hurt...