Number theory

Can you produce convincing arguments that a selection of statements about numbers are true?

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.

Which numbers can we write as a sum of square numbers?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.