A059106 Number of solutions to Nickerson variant of Langford (or Langford-Skolem) problem.

1, 0, 0, 3, 5, 0, 0, 252, 1328, 0, 0, 227968, 1520280, 0, 0, 700078384, 6124491248, 0, 0, 5717789399488, 61782464083584, 0, 0, 102388058845620672, 1317281759888482688, 0, 0, 3532373626038214732032, 52717585747603598276736, 0, 0

Offset: 1

Author

N. J. A. Sloane, Feb 14 2001

Keywords

• nonn
• nice
• hard
• more

Comments

How many ways are of arranging the numbers 1,1,2,2,3,3,...,n,n so that there are zero numbers between the two 1's, one number between the two 2's, ..., n-1 numbers between the two n's?

For n > 1, a(n) = A004075(n)/2 because A004075 also counts reflected solutions. - Martin Fuller, Mar 08 2007

Because of symmetry, is a(5) = 5 the largest prime in this sequence? - Jonathan Vos Post, Apr 02 2011

Examples

			For n=4 the a(4)=3 solutions, up to reversal of the order, are:
1 1 3 4 2 3 2 4
1 1 4 2 3 2 4 3
2 3 2 4 3 1 1 4
From _Gheorghe Coserea_, Aug 26 2017: (Start)
For n=5 the a(5)=5 solutions, up to reversal of the order, are:
1 1 3 4 5 3 2 4 2 5
1 1 5 2 4 2 3 5 4 3
2 3 2 5 3 4 1 1 5 4
2 4 2 3 5 4 3 1 1 5
3 5 2 3 2 4 5 1 1 4
(End)
		

Links

• Ali Assarpour, Amotz Bar-Noy, Ou Liuo, Counting the Number of Langford Skolem Pairings, arXiv:1507.00315 [cs.DM], 2015.
• Gheorghe Coserea, Solutions for n=8.
• Gheorghe Coserea, Solutions for n=9.
• J. E. Miller, Langford's Problem
• R. S. Nickerson and D. C. B. Marsh, E1845: A variant of Langford's Problem, American Math. Monthly, 1967, 74, 591-595.
• Zan Pan, Conjectures on the number of Langford sequences, (2021).

Crossrefs

Cf. A014552, A050998, A059107, A059108.

Cf. A004075, A268535.

Extensions

a(20)-a(23) from Mike Godfrey (m.godfrey(AT)umist.ac.uk), Mar 14 2002

Extended using results from the Assarpour et al. (2015) paper by N. J. A. Sloane, Feb 22 2016 at the suggestion of William Rex Marshall

a(28)-a(31) from Assarpour et al. (2015), added by Max Alekseyev, Sep 24 2023