Mathematical Description

For both Skolem and Langford sequences if n is divisible by 4 then there are solutions. Langford sequences also appear to have solutions if after dividing n by 4 there is a remainder of 3 and Skolem sequences have solutions if the remainder is 1.

Definition: Langford and Skolem sequences can both be generalized as follows:
Two of each number S = [1, 2...n] with the property: y_i = x_i + t_i for i = 1, 2...n. Langford sequences: y_i = x_i + i + 1, S = [1, 2...n] and i = 2...n + 1
Skolem sequences: y_i = x_i + i, S = [1, 2...n] and i = 1, 2...n

In both cases we would say the Langford or Skolem sequence is order n and defect 1, meaning we start with 1 and the largest number is n.

We can even further generalize Langford and Skolem sequences in a couple ways.
For instance, hooked Skolem and Langford sequences allows zeroes or empty spaces: 41134030
Additionally, we can have a Skolem or Langford sequence that contains only a subset of 1, 2...n such as: 75311357

Skolem squares use both these properties, allowing 0s and a subset of 1, 2..n in each row and column of the square.