

A014552


Number of solutions to Langford (or LangfordSkolem) problem (up to reversal of the order).


19



0, 0, 1, 1, 0, 0, 26, 150, 0, 0, 17792, 108144, 0, 0, 39809640, 326721800, 0, 0, 256814891280, 2636337861200, 0, 0, 3799455942515488, 46845158056515936, 0, 0, 111683611098764903232, 1607383260609382393152, 0, 0
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OFFSET

1,7


COMMENTS

These are also called Langford pairings.
2*a(n) = A176127(n) gives the number of ways of arranging the numbers 1,1,2,2,...,n,n so that there is one number between the two 1's, two numbers between the two 2's, ..., n numbers between the two n's.
a(n) > 0 iff n == 0 or 3 (mod 4).


REFERENCES

Jaromir Abrham, "Exponential lower bounds for the numbers of Skolem and extremal Langford sequences," Ars Combinatoria 22 (1986), 187198.
M. Gardner, Mathematical Magic Show, New York: Vintage, pp. 70 and 7778, 1978.
M. Gardner, Mathematical Magic Show, Revised edition published by Math. Assoc. Amer. in 1989. Contains a postscript on pp. 283284 devoted to a discussion of early computations of the number of Langford sequences.
R. K. Guy, The unity of combinatorics, Proc. 25th Iranian Math. Conf, Tehran, (1994), Math. Appl 329 129159, Kluwer Dordrecht 1995, Math. Rev. 96k:05001.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 2.
M. Krajecki, Christophe Jaillet and Alain Bui, "Parallel tree search for combinatorial problems: A comparative study between OpenMP and MPI," Studia Informatica Universalis 4 (2005), 151190.
Roselle, David P. Distributions of integers into stuples with given differences. Proceedings of the Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1971), pp. 3142. Dept. Comput. Sci., Univ. Manitoba, Winnipeg, Man., 1971. MR0335429 (49 #211).  From N. J. A. Sloane, Jun 05 2012


LINKS



FORMULA



EXAMPLE

Solutions for n=3 and 4: 312132 and 41312432.
Solution for n=16: 16, 14, 12, 10, 13, 5, 6, 4, 15, 11, 9, 5, 4, 6, 10, 12, 14, 16, 13, 8, 9, 11, 7, 1, 15, 1, 2, 3, 8, 2, 7, 3.


CROSSREFS

See A050998 for further examples of solutions.
If the zeros are omitted we get A192289.


KEYWORD

nonn,hard,nice,more


AUTHOR



EXTENSIONS

a(20) from Ron van Bruchem and Mike Godfrey, Feb 18 2002
a(21)a(23) sent by John E. Miller (john@timehaven.us) and Pab Ter (pabrlos(AT)yahoo.com), May 26 2004. These values were found by a team at Université de Reims ChampagneArdenne, headed by Michael Krajecki, using over 50 processors for 4 days.
a(24)=46845158056515936 was computed circa Apr 15 2005 by the Krajecki team.  Don Knuth, Feb 03 2007


STATUS

approved



