OlympiadBench / 2319 / A Skolem sequence of order n is a sequence …

A Skolem sequence of order n is a sequence (s₁,s₂,…,s_(2n)) of 2n integers satisfying the conditions:

i) for every k in {1,2,3,…,n}, there exist exactly two elements s_(i) and s_(j) with s_(i)=s_(j)=k, and

ii) if s_(i)=s_(j)=k with i<j, then j−i=k.


For example, (4,2,3,2,4,3,1,1) is a Skolem sequence of order 4.
Determine, with justification, all Skolem sequences of order 9 which satisfy all of the following three conditions:
I) s₃=1,

II) s₁₈=8, and

III) between any two equal even integers, there is exactly one odd integer.

A Skolem sequence of order $n$ is a sequence $\left(s_{1}, s_{2}, \ldots, s_{2 n}\right)$ of $2 n$ integers satisfying the conditions:

i) for every $k$ in $\{1,2,3, \ldots, n\}$, there exist exactly two elements $s_{i}$ and $s_{j}$ with $s_{i}=s_{j}=k$, and

ii) if $s_{i}=s_{j}=k$ with $i<j$, then $j-i=k$.


For example, $(4,2,3,2,4,3,1,1)$ is a Skolem sequence of order 4.
Determine, with justification, all Skolem sequences of order 9 which satisfy all of the following three conditions:
I) $s_{3}=1$,

II) $s_{18}=8$, and

III) between any two equal even integers, there is exactly one odd integer.