A059106 - OEIS

%I #58 Sep 24 2023 10:56:26

%S 1,0,0,3,5,0,0,252,1328,0,0,227968,1520280,0,0,700078384,6124491248,0,

%T 0,5717789399488,61782464083584,0,0,102388058845620672,

%U 1317281759888482688,0,0,3532373626038214732032,52717585747603598276736,0,0

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

%C 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?

%C For n > 1, a(n) = A004075(n)/2 because A004075 also counts reflected solutions. - _Martin Fuller_, Mar 08 2007

%C Because of symmetry, is a(5) = 5 the largest prime in this sequence? - _Jonathan Vos Post_, Apr 02 2011

%H Ali Assarpour, Amotz Bar-Noy, Ou Liuo, <a href="http://arxiv.org/abs/1507.00315">Counting the Number of Langford Skolem Pairings</a>, arXiv:1507.00315 [cs.DM], 2015.

%H Gheorghe Coserea, <a href="/A059106/a059106.txt">Solutions for n=8</a>.

%H Gheorghe Coserea, <a href="/A059106/a059106_1.txt">Solutions for n=9</a>.

%H J. E. Miller, <a href="http://dialectrix.com/langford.html">Langford's Problem</a>

%H R. S. Nickerson and D. C. B. Marsh, <a href="http://www.jstor.org/stable/2314911">E1845: A variant of Langford's Problem</a>, American Math. Monthly, 1967, 74, 591-595.

%H Zan Pan, <a href="https://eprint.panzan.me/articles/langford.pdf">Conjectures on the number of Langford sequences</a>, (2021).

%e For n=4 the a(4)=3 solutions, up to reversal of the order, are:

%e 1 1 3 4 2 3 2 4

%e 1 1 4 2 3 2 4 3

%e 2 3 2 4 3 1 1 4

%e From _Gheorghe Coserea_, Aug 26 2017: (Start)

%e For n=5 the a(5)=5 solutions, up to reversal of the order, are:

%e 1 1 3 4 5 3 2 4 2 5

%e 1 1 5 2 4 2 3 5 4 3

%e 2 3 2 5 3 4 1 1 5 4

%e 2 4 2 3 5 4 3 1 1 5

%e 3 5 2 3 2 4 5 1 1 4

%e (End)

%Y Cf. A014552, A050998, A059107, A059108.

%Y Cf. A004075, A268535.

%K nonn,nice,hard,more

%O 1,4

%A _N. J. A. Sloane_, Feb 14 2001

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

%E 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_

%E a(28)-a(31) from Assarpour et al. (2015), added by _Max Alekseyev_, Sep 24 2023