Problem A1

Let n be a fixed positive integer. How many ways are there to write n as a sum of positive integers,

n = a₁ + a₂ + ··· + a_k,

with k an arbitrary positive integer and a₁ £ a₂ £ ··· £ a_k £ a₁ + 1? For example, with n = 4, there are four ways: 4, 2 + 2, 1 + 1 + 2, 1 + 1 + 1 + 1.