Necessary Conditions for Existence of Skolem Sequence

Theorem
A Skolem sequence of order $$n$$ can only exist if $$n\equiv 0,1 (\bmod{4})$$.

Proof
Let $$S$$ be a Skolem sequence of order $$n$$ and let $$a_i$$ and $$b_i$$ be the positions of the first and second occurrences respectively of the integer $$i$$ in $$S$$, where $$1 \le i \le n$$.

We can thus conclude that $$b_i - a_i = i,$$ for each $$i$$ from 1 to $$n$$.

Summing both sides of this equation we obtain:
 * $$\sum_i^n b_i - \sum_i^n a_i = \sum_i^n i = \frac{n(n + 1)}{2}$$

Now the $$a_i$$ and the $$b_i$$ represent the positions in the sequence from $$1$$ to $$2n$$, so we can see that:
 * $$\sum_i^n b_i + \sum_i^n a_i = \frac{2n(2n + 1)}{2} = n(2n + 1)$$

Summing the previous two equations we obtain the identity:
 * $$\sum_i^n b_i = \frac{n(5n + 3)}{4}$$

The left hand side of this last equality is a sum of positions and thus must be integer.

We conclude that the right hand side must also be an integer, which occurs exactly when: $$n \equiv 0, 1 (\bmod{4})$$.