Problem $2(15$ points$)$: A sequence of integers is called a supercombination for $\backslash ($ n $\backslash )$ if for any pair of integers between $1$ and $\backslash ($ n $\backslash )($inclusive$),$ that pair appears as two consecutive values in the sequence $($in some order$).$ For example, the following is a supercombination for $4,$ with all the pairs highlighted.

Note that there are exactly $\backslash ($ n$($n$-1)/2\backslash )$ pairs of integers between $1$ and $\backslash ($ n $\backslash ).$ As such, any supercombination must have at least $\backslash ($ n$($n$-1)/2+1\backslash )$ elements in it$.$

We say that a supercombination is optimal if it has that exact length. For example, the above supercombination is not optimal, as $\backslash (8>4\backslash$cdot $3/2+1=7\backslash ).$

For which values of $\backslash ($ n $\backslash )$ do optimal supercombinations exist?