Ordered sums

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

Problem

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. For example, a(4) = 5 because:

4 =2 + 2
 2 + 1 + 1
 1 + 2 + 1
 1 + 1 + 2
 1 + 1 + 1 + 1.

Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1.

(i)Calculate a(n) and b(n) for n ≤ 8. What do you notice about these sequences?
(ii)Find a relation between a(p) and b(q).
(iii)Prove your conjectures.