My son is very interested nowadays in looking for patterns.
He started with simple patterns like ABCABC …, then developed to complex ones like AABCCCAABCCC, (where A, B & C can stand for colors, shapes, letters, numbers, objects, animals, etc… ) or even combinations of these. Then he started applying the concept to numerics like 1,3,5,7,… or 11,22,33,… or even 120,230,340,…
It is an interesting game to try to solve the rule/formula and guess the whole series.
For example: The series 3,5,7, … is based on the formula: 2n+1 where n=1,2,3,…
More info about sequences can be found at
http://www.mathsisfun.com/algebra/sequences-series.html
More series/patterns examples are most welcome.
Rima