Hint to puzzle 146: Odds and evens

1. How many different possible games are there?

   Let g(n) denote the number of different possible games with upper bound n.  Express g(n) in terms of g(n − a) and g(n − b), where a and b are constants to be determined.

2. How many different possible games of length k are there?

   Let {1 a₁ < a₂ < ... < a_k = n} denote a game of length k.  Consider the sequence defined by b_i = a_i − i + 1.