A004075 Number of Skolem sequences of order n.

1, 0, 0, 6, 10, 0, 0, 504, 2656, 0, 0, 455936, 3040560, 0, 0, 1400156768, 12248982496, 0, 0, 11435578798976, 123564928167168, 0, 0, 204776117691241344, 2634563519776965376, 0, 0, 7064747252076429464064, 105435171495207196553472, 0, 0

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more

Comments

Number of permutations of the multiset {1,1,2,2,...,n,n} such that the distance between the elements i equals i for every i=1,2,...,n.

Number of super perfect rhythmic tilings of [0,2n-1] with pairs. See A285698 and A285527 for the definition and tilings of triples and quadruples. - Tony Reix, Apr 25 2017

References

• CRC Handbook of Combinatorial Designs, 1996, p. 460.

Links

• Ali Assarpour, Amotz Bar-Noy, Ou Liuo, Counting the Number of Langford Skolem Pairings, arXiv:1507.00315 [cs.DM], 2015.
• S. Burrill and L. Yen, Constructing Skolem sequences via generating trees, arXiv preprint arXiv:1301.6424 [math.CO], 2013.
• J. E. Miller, Langford's Problem
• G. Nordh, Perfect Skolem sequences, arXiv:math/0506155 [math.CO], 2005.

Crossrefs

Cf. A014552, A059106, A176127, A268537.

Programs

Mathematica

(* Program not suitable to compute a large number of terms. *)
iter[n_] := Sequence @@ Table[{x[i], {-1, 1}}, {i, 1, 2n}];
a[n_] := 1/2^(2n) Sum[Product[x[i], {i, 1, 2n}] Product[Sum[x[k] x[k+i], {k, 1, 2n-i}], {i, 1, n}], iter[n] // Evaluate];
Table[Print[a[n]]; a[n], {n, 1, 10}] (* Jean-François Alcover, Sep 29 2018, from formula in Assarpour et al. *)

Formula

For n > 1, a(n) = A059106(n)*2 because A059106 ignores reflected solutions. - Martin Fuller, Mar 08 2007

Extensions

More terms (via A059106) from Martin Fuller, Mar 08 2007

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