The Fibonnaci sequence occurs so frequently because it is the solution of the simplest of all difference relations. It is instructive to view it in this way and perhaps to introduce the idea of difference equations with this familiar example.

Proving these results calls for considering whether or not other terms in the sequences, apart from those in the recognized patterns, can also be multiples of 2 or 3 respectively in the two cases. Are the conditions necessary as well as sufficient?