Challenge Level

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. |