Cool problem

Number Theory Level pending

Let us call a sum of integers cool if the first and last terms are 1 and each term differs from its neighbours by at most 1. For example, the sum 1 +2+ 3+4 +3+ 2 + 3+3 + 3+2 + 3+3 + 2+1 is cool. How many terms does it take to write 2008 as a cool sum if we use no more terms than necessary?

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