Arrange copies of the
digits 1, ...,
such that there is one digit between the 1s, two digits between the 2s, etc. For
example, the unique (modulo reversal) solution is 231213, and the unique (again modulo reversal)
solution is 23421314.

Davies (1959) showed that solutions to Langford's problem exist iff (cf. Assarpour et al.
2017), so the next solutions occur for . There are 26 of these, as exhibited by Lloyd (1971). In
lexicographically smallest order (i.e., small digits come first), the first few Langford
sequences are 231213, 23421314, 14156742352637, 14167345236275, 15146735423627, ...
(OEIS A050998).

The number of solutions for , 4, 5, ... (modulo reversal of the digits) are 1, 1, 0,
0, 26, 150, 0, 0, 17792, 108144, ... (OEIS A014552).
No formula is known for the number of solutions of a given order , but Assarpour et al. (2017) computed their counts
up to ,
making
the smallest unknown case (since there are no Langford sequences of order 29 or 30).