# Smooth Partitons

A smooth partition of the integer $$n$$ is a set of positive integers $$a_1, a_2, \ldots a_k$$ such that
1. $$k$$ is a positive integer,
2. $$a_1 \leq a_2 \leq \cdots \leq a_k,$$
3. $$\sum_{i=1}^{k} a_i = n,$$ and
4. $$a_k - a_1 \leq 1.$$
Determine how many smooth partitions there are of the integer $$250.$$

Details and assumptions

Note: The definition given above is not standard terminology.

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