Langford's Problem Continued

Can we go further than 312132? Can we make Langford sequences with numbers 1-4, 1-5 and so on. Langford only found sequences for numbers 1 through 4, 7, 8, 11, 12 and 15 and was unable to find sequences for 5, 6, 9 and 10. Langford's problem thus asks the question: for which n can Langford sequences be constructed?

Examples of a few Langford sequences for n = 3, 4, 7 and 8:

312132, 41312432, 45671415362732, 4567841516372832

Skolem sequences are similar, indexing at 0 instead of 1 for the number of spaces separating each pair. Examples of a few Skolem sequences for n = 4, 5, 8 and 9:

41134232, 5113453242, 8113673485647252, 478349357682529611