Find a recurrence relation for the sequence $u_n=$ number of nonnegative integral solutions of $2a+5b=n$.

Question

Find a recurrence for the sequence $u_n=$ number of nonnegative integral solutions of $$2a+5b=n.$$

I think I can use a generating function, but I'm a bit confused at this point. Is anyone is able to give me a hint? I would like to solve the problem myself, so of emblem, I would ask you not to solve the problem.

Answer 1

Hints:

• Integer solutions to $2a+5b=n$ may be related to integer solutions to $2(a-1)+5b=n-2$ and to $2a+5(b-1)=n-5$
• So you might expect the recurrence to involve $u_{n-2}$ and $u_{n-5}$, and possibly $u_{n-7}$
• You could consider what $u_n-u_{n-2}$ and $u_n-u_{n-5}$ each count