Some Mathematica code for counting number of solutions by brute force (and links to some previous solutions).

File	Type	Notes
Langford.aim	AIMMS+CP
langford.lp4	ASP Gringo 4
langford_bp.pl	B-Prolog
langford.co	Comet
langford.ecl	ECLiPSe CLP
Langford_jacop.scala	JaCoP/Scala
langford2.mzn	MiniZinc
langford.cs	or-tools/C#
langford_ortools.py	or-tools/Python
Langford_oscar.scala	OscaR
langford.pi	Picat
ROD.pl	pl	A Perl program written by John Miller based on a pattern found by Roy O. Davies that generates a single Langford sequence (for large n). Counting the number of sequences is likely to be intractable.
langford_sicstus.pl	SICStus Prolog