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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. |